Sudoku, the Golden Ratio, Determinants, Eigenvectors and Matrices - PDF

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					      Sudoku, the Golden Ratio, Determinants, Eigenvectors and
                          Matrices In Z

                                               Merciadri Luca⋆
                                      Luca.Merciadri@student.ulg.ac.be


        Abstract. The Sudoku game has gained much interest for a dozen years. It is now put in various mag-
        azines and mathematical research is more and more interested in it. This document aims at providing
        some newer pieces of information about the mathematical properties of the Sudoku, but not according
        to graphs’ theory. We do not speak about the “minimum number of clues,” but about (erroneous or
        not) Sudokus’ matrix interpretation: general properties of its determinant, in relation to eigenvalues
        and eigenvectors; transpose; neither symmetric nor antisymmetric character; non-Hermitian character;
        non-normal character and non-orthogonal character; trace and antitrace. A part is also devoted to the
        proposal of Prof. Steven Leon, which states that, given a 9×9 Sudoku matrix, the maximum eigenvalue
        of a given 3 × 3 submatrix extracted from it, equals 22. Last sections give results about Numerical
        Analysis of Sudoku matrices and the representation of a Sudoku when its matrix is taken modulo 2,
        and then considered as the adjacency matrix of a graph.


    Keywords: Sudoku, Determinants, Matrix, Matrices, Eigenvalues, Eigenvectors.
REVISION 1: 08/05/2010


1     Introduction
The Sudoku game has gained much interest for a dozen years. It is now put in various magazines and
mathematical research is more and more interested in it. This document aims at providing some newer
pieces of information about the mathematical properties of the Sudoku, but not according to graphs’ theory.
We do not speak about the “minimum number of clues,” but about (erroneous or not) Sudokus’ matrix
interpretation: general properties of its determinant, in relation to eigenvalues and eigenvectors; transpose;
neither symmetric nor antisymmetric character; non-Hermitian character; non-normal character and non-
orthogonal character. A part is also devoted to the proposal of Prof. Steven Leon, which states that, given
a 9 × 9 Sudoku matrix, the maximum eigenvalue of a given 3 × 3 submatrix extracted from it, equals 22.


2     Conventions
Through this paper, we use the following conventions:
 – S denotes the set of all the (solved) Sudokus. We have |S| ≈ 6.771 · 1021 , according to [3],
 – Si,j denotes the element at the ith row and the jth column of the general S Sudoku square matrix. We
   shall use 1 ≤ i ≤ n, 1 ≤ j ≤ n, where n denotes the number of rows (or columns, as it is always a square
   matrix) of the Sudoku (we always consider n ≥ 4). S has thus the following form:
                                                                    
                                                S1,1 S1,2 · · · S1,n
                                               S2,1 S2,2 · · · S2,n 
                                                                    
                                          S= .        . ..      . ,
                                               . .    .
                                                       .      . .. 
                                                Sn,1 Sn,2 · · · Sn,n
⋆
    The author is thankful to Prof. Steven Leon, University of Massachussets, Dartmouth, for his idea of the maximum
    eigenvalue in a submatrix of a Sudoku 9 × 9 matrix.
        n
 –      j=1  Si,j = n(n+1) , 1 ≤ i ≤ n (i being free in this interval) denotes the sum of all the elements on one
                       2
      line i of S. Evidently,
                                        n                       n
                                                  n(n + 1)                n(n + 1)
                                           Si,j =           =      Si,j =          ,
                                      j=1
                                                      2        i=1
                                                                             2

   because of the fundamental properties of the Sudoku,
 – Z∗ denotes the set of the integers without 0; that is, Z \ {0},
 – C is the transpose of the matrix C,
 – C ∗ is the adjoint of the matrix C, i.e. C,
                n                       n
 – Tr(C) = i=1 Ci,i , antiTr(C) = i=1 Ci,n−i+1 ,
 – M i,j is the algebraic minor of the element at the ith row and the jth column of a given matrix,
 – cofact(Ci,j ) = (−1)i+j · M i,j ,
 – The dimension of a Sudoku matrix is expressed as a product n × n, where n denotes its dimension, and
   n2 the number of elements it is filled with,
 – When taking matrices with a n > 9 (i.e. 16 × 16 or 25 × 25), we assume there exists a sufficiently
   complete alphabet A which has n distinct symbols. For example, when speaking about 16 × 16 Sudokus,
   “A” is used for “10,” “B” is used for “11,” . . . , until “G” is used for “16.”

Evidently, we assume the reader is familiar with the notions of Sudoku, Latin Square, and related concepts.


3     Determinants

3.1     General Case

As S is a square matrix, one can wonder about the value of det(S), for a given n. Based on many experiments,
det S has the following properties:

 – It is in Z∗ or in Z,
 – The assertion
                                  (det(S) (mod 2) = 0) ∨ (det(S)      (mod 2) = 0)                           (1)
      is always true, but det S is sometimes even, and sometimes odd.


3.2     Why it Sometimes Does Not Equal Zero

If det(S) = 0, S −1 cannot exist, and 0 must be an eigenvalue of the spectrum of S. Thus, proving that, for a
given n, (det(S) = 0) is sometimes a true assertion is the same as proving that 0 is sometimes an eigenvalue
of the spectrum of S. However, proving that 0 is sometimes an eigenvalue of the spectrum of S would be
less straightforward.
    The rows of the Sudoku are sometimes linearly dependent, and we can only assume that (det(S) = 0) is
sometimes a true assertion.

Theorem 1 (Determinant of S sometimes equals 0). The determinant of the S matrix, det S, formed
by the elements of a complete Sudoku, can equal 0.                                            ⊔
                                                                                              ⊓

Proof. If det(S) = 0, the rows of S are linearly dependent. We prove first that there exists at least a situation
where det(S) = 0, S still being a valid Sudoku matrix. It is shown above for at least two rows (it is the
same for columns, just transpose). There are n! possible orders of the elements of a row of S. From these
possibilities,
 – Let’s take
                                                 947258136
                                                 123467985
                                                 658193724
                                                 895642371
                                        det(B) = 7 6 4 9 3 1 2 5 8 = 0,
                                                 312875469
                                                 489726513
                                                 236519847
                                                 571384692

 – It is shown in [2] that

       abcd   abdc
       cdab   cdba
            =      = a4 + b4 + c4 + d4 − 2a2 b2 − 2a2 c2 − 2b2 c2 − 2a2 d2 − 2b2 d2 − 2c2 d2 + 8abcd,
       dcba   bacd
       badc   dcab

   which is logical, as there is an even number of permutations of rows (and columns) between the first
   and the second matrix. In the standard case, i.e. if (a, b, c, d) are mapped bijectively to (1, 2, 3, 4), we
   have
               a4 + b4 + c4 + d4 − 2a2 b2 − 2a2 c2 − 2b2 c2 − 2a2 d2 − 2b2 d2 − 2c2 d2 + 8abcd = 0.

   Note that
                                               673514982
                                               921368754
                                               584792136
                                               896237415
                                               2 1 5 9 4 6 3 7 8 = 0.
                                               347185269
                                               452871693
                                               169453827
                                               738629541
   This result will be used later.
                                                                                                             ⊔
                                                                                                             ⊓

Remark 1 (Possible configurations). Given a matrix S, [at least] two rows in S will never be formed by the
same elements in the same order than another row. If this happens, S is not a valid Sudoku anymore, as it
leads to a part of S (the example is for the rows of S, but it is as trivial for the columns of S) like
                                                               
                                             ··· ··· ··· ··· ···
                                            1 2 3 ··· n 
                                                               
                                            1 2 3 ··· n ,
                                             ··· ··· ··· ··· ···

where 1, 2, · · · , n can be in a different order. This would also be shown by the determinant of a submatrix
of
                                                k k + 1 k + 2 ··· n
                                                                    ,
                                                k k + 1 k + 2 ··· n

which always equals 0 (thus leading to a rank ρ less than 2),
    Beginning by the end, i.e. using the following configuration:
                                                                
                                            ··· ··· ··· ··· ···
                                          1     2     3 ··· n 
                                                                
                                           n n − 1 n − 2 ··· 1 
                                            ··· ··· ··· ··· ···

can only lead to linearly independent rows. The rank ρ of a matrix A being defined as the biggest dimension
of the square submatrices of = 0 det extracted from A, we know that
                                                             
                                         ··· ··· ··· ··· ···
                                       1    2     3 · · · n 
                                    ρ                        
                                       n n − 1 n − 2 · · · 1  = 2,
                                         ··· ··· ··· ··· ···

under some given conditions which will be determined now. For k ∈ {1, · · · , n − 1}, we have

                         k k+1
                               = k(n − k) − n(k + 1) = kn − k 2 − kn − n = −k 2 − n;
                         nn−k

That is, −k 2 − n ≤ 2 iff −n ≤ 2 + k 2 , leading to n ≥ −2 − k 2 . As −2 − k 2 = −3 iff k = 1 and
                                          k=(n−1)
                               −2 − k 2     =       −2 − (n − 1)2 = −n2 + 2n + 3,

we must ask
                                                n ≥ −n2 + 2n + 3
                                                                     ,
                                                n ≥ −3
which is equivalent to                                   √         √
                                      n    ∈ − ∞, 1−2        5   1+ 5
                                                                   2 , +∞
                                                                            ,
                                      n    ≥ −3
thus giving                                              √         √
                                      n    ∈ − ∞, 1−2        5   1+ 5
                                                                   2 , +∞
                                                                            ,
                                      n    ≥ −3
and                                                      √
                                                      1+ 5
                                                n∈         , +∞ ,
                                                        2
which is always the case, as
                                  √
                               1+ 5
                                    ≃ 1.6180339887498948482045868343656,
                                 2
and the dimension of a Sudoku is always the square of a number. Thus, the first interesting Sudoku’s
dimension would be 4 = 22 (thus greater than the golden ratio).                                  ⊔
                                                                                                 ⊓


4     Erroneous Sudokus

Theorem 2 (Finding if a Sudoku matrix is erroneous or not). There is no way to be sure about the
correctness of a Sudoku matrix by only computing its determinant.                              ⊔
                                                                                               ⊓
Proof. An erroneous Sudoku can have a determinant of zero, but it is not always the case. Furthermore,
having a determinant of zero cannot even be a sufficient condition to be an erroneous Sudoku matrix, as
there are correct Sudoku matrices which have a determinant of 0.
   Let’s take two erroneous Sudokus:
 1. The first will verify,
                                         ··· ··· ··· ··· ··· ··· ··· ··· ···
                                          1 2 3 4 5 6 7 8 9
                                                                             = 0,
                                          1 2 6 5 4 3 7 9 8
                                         ··· ··· ··· ··· ··· ··· ··· ··· ···
    assuming the · · · can be filled correctly to make no other mistake. At least a given Sudoku can thus
    have a non-zero determinant,
 2. The second is “more” erroneous:
                                         ··· ··· ··· ··· ··· ··· ··· ··· ···
                                          1 2 3 4 5 6 7 8 9
                                                                             = 0,
                                          1 2 3 4 5 6 7 8 9
                                         ··· ··· ··· ··· ··· ··· ··· ··· ···

      which is very clear, as
                                                 123456789
                                            ρ                           = 1.
                                                 123456789
      We have also seen before that
       abcd   abdc
       cdab   cdba
            =      = a4 + b4 + c4 + d4 − 2a2 b2 − 2a2 c2 − 2b2 c2 − 2a2 d2 − 2b2 d2 − 2c2 d2 + 8abcd,
       dcba   bacd
       badc   dcab

which equals 0 iff (a, b, c, d) are mapped bijectively to (1, 2, 3, 4), leading however to a correct Sudoku.    ⊔
                                                                                                               ⊓


5      Sudoku Eigenvalues
5.1     General Case
Let’s take the eigenvalues of S; that is, let’s compute det(S −λI). Depending on the dimension of S, denoted
by n, det(S − λI) will be a polynom of a different order: this order will always be of n. That is,
                                                                               n
                                det(S − λI) = αλn + βλn−1 + · · · + ζλ0 =            αi λi                    (2)
                                                                               i=0

for a general S matrix of dimension n.

5.2     Example: Dimension Nine
As a 9 × 9 Sudoku is also a magic square, det(S − λI) = 0 for at least a λi = 45 (also found in [3]). The
spectrum of S (restricted to a real spectrum, as there is no order in C, if it is composed of some complex
values, and this is a situation which is possible and current), composed of the λi , verifies thus

                                                   max λi = 45.
                                                1 ≤ i ≤ n,
                                                  λi ∈ R
Equation 2 can be rewritten (for a 9 × 9 Sudoku) as
                                                                             n−1
                                          det(S − λI) = C(λ − 45)                  (λ − λi ),              (3)
                                                                             i=1

C being a constant in Z∗ , as we are in a sufficiently big structure to have n roots, counted with their
multiplicities.

Why 45? One can notice easily two facts:
 – 9 × 5 = 45,
                             9(9+1)       90
 – 1 + 2 + 3 + ··· + 9 =        2     =    2   = 45.
Thus, the dominant eigenvalue in the spectrum of S, with a size of 9, equals the sum of all the elements of
a given line (the first fact cannot be extended to other dimensions).
    It is said in [3] that the determinant of a Sudoku 9 × 9 is always divisible by 45, and it is even the case
if at least one eigenvalue is in C. It can be seen very easily, as
                                                                       λ=0
                                                    det(S − λI) = det(S).                                  (4)
We then have
                                                          n−1
                                                                          λ=0
                                                D(45)           (λ − λi ) = det(S),                        (5)
                                                          i=1
the 45 factor multiplying every other λi , with D = −C, |D| = |C|.

5.3   Return to General Case
Theorem 3 (Perron). If all of the entries of a matrix are positive, then the matrix has a dominant
eigenvalue that is real and has multiplicity 1.                                                 ⊔
                                                                                                ⊓
The proof of this theorem is not given here.
Theorem 4 (Max eigenvalue of a square and positive matrix). The dominant eigenvalue of any
square and positive matrix where each row and column have the same sum, will equal that sum. ⊔
                                                                                             ⊓
Proof. Easily stated by using Theorem 3.                                                                     ⊔
                                                                                                             ⊓
Corollary 1 (Eigenvalues of S). A Sudoku matrix S with a dimension of n always verifies
                                                                   n
                                                                                 n(n + 1)
                                             max λi =                   Si,j =            .
                                          1 ≤ i ≤ n,              j=1
                                                                                    2
                                            λi ∈ R
                                                                                                             ⊔
                                                                                                             ⊓
Proof. Trivial using Theorem 4.                                                                              ⊔
                                                                                                             ⊓
Corollary 2. A Sudoku square matrix S, of dimension n, and of determinant det(S), always verifies
                                                          
                                                                
                                          det(S) mod  max λi  = 0,
                                                     1 ≤ i ≤ n, 
                                                        λi ∈ R
which is equivalent to ask
                                                                       n(n + 1)
                                               det(Si )    mod                       = 0.
                                                                          2
                                                                                                             ⊔
                                                                                                             ⊓
Proof. Trivial, as
                                               det(S − λI) = det(S)
when λ = 0, and using Corollary 2.                                                                            ⊔
                                                                                                              ⊓
Example 1 (Divisibility of the determinant of S if it represents a 4 × 4 Sudoku). Whatever the 4 × 4 Sudoku,
its determinant is always divisible by 4(4+1) = 10. Evidently, it can be divisible by 45, for example, but it
                                          2
might not always be the case.                                                                              ⊔
                                                                                                           ⊓

5.4     Subsudoku Eigenvalues
Prof. Steven Leon from the University of Massachussets Dartmouth said in an e-mail conversation we had,
that
for each of the 9 submatrices of a (9 × 9) Sudoku matrix, the absolute value of any (real) eigenvalue of the
                                    submatrix is less than or equal to 22.
This statement is generalized in Theorem 6, but we first have to speak about the submatrices which can be
extracted from a n × n given Sudoku matrix. They are never Sudoku ones.
Theorem 5 (Submatrices of a Sudoku matrix are not Sudoku matrices anymore). The subma-
trices of a given Sudoku matrix are never Sudoku matrices.                                                  ⊔
                                                                                                            ⊓
                                                                                                    √     √
Proof. Let’s take a n × n Sudoku matrix, n being the square of an integer. Its submatrices will be n × n.
              √                                                                           √
The number n is √   always in Z. (If it was not, the submatrix would not be a matrix!) If n is not the square
                          √
of an integer, the n × n submatrices cannot be, by definition, Sudoku ones (an example is the 9 × 9
                                                                            √
Sudoku case: 3 × 3 submatrices are not Sudoku matrices anymore). If n is the square of an integer (e.g.
                                                                                   √     √
16 × 16 Sudoku matrices, resulting in 4 × 4 submatrices), it won’t prevent the n × n submatrices from
being partially filled with elements in {1, · · · , n}, thus leading to a non-Sudoku matrix, by definition.   ⊔
                                                                                                            ⊓
      We now give the important theorem about the maximum eigenvalue of a given Sudoku’s submatrix.
Theorem 6 (Maximum eigenvalue of a Sudoku submatrix). If S is the matrix of a n × n Sudoku,
with n being a square number (i.e. there exists one a ∈ N∗ so that a × a = n), the maximum (bounding)
                                      ′                                                ′
eigenvalue of one of its submatrices Sk , k ∈ {1, · · · , n}, is, if we consider that Sk = S ′ :
                                                                                 
                             
                                                                                     
                                                                                      
                             
                                                                                   
                             
                             
                                             √
                                                           
                                                                            √
                                                                                     
                                                                                      
                               
                                               n
                                                     ′ 
                                                                              n
                                                                                     
                          min  max               Si,j  ,  max√               Si,j  .
                                                                                  ′
                                                                                                  (6)
                             i∈{1,··· ,√n}
                                                         j∈{1,··· , n}
                                                             
                                                                                     
                                                                                     
                             
                                            j=1                          i=1        
                                                                                      
                             
                                                                                     
                                                                                      
                                           = S   ∞                    = S    1



This is an upper bound, and there is not necessarily an eigenvalue of these submatrices which equals this
value.                                                                                                 ⊔
                                                                                                       ⊓
                                                       ′
Proof. If λ1 , · · · , λp are the eigenvalues of any Sk submatrix taken from S, it is known (this is property of
matrix norms) that
                                                              ′
                                                     |λm | ≤ Sk ,
                   ′
whatever the Sk (k having the same value in this proof, from the beginning to the end) and the m ∈
                       ′                 ′                                                      ′      ′
{1, · · · , p}, where Sk is the norm of Sk . As, if, to simplify writings, we consider locally Si,j = Ski,j ,
                                                                               
                                          √                             √
                                                                                      
                                            n                             n
                                                                                      
                                                     
                             max               ′
                                               Si,j    and  max√
                                                            j∈{1,··· , n}
                                                                                ′
                                                                               Si,j   
                                                                                      
                            i∈{1,··· ,√n}                                          
                                          j=1                            i=1

                                     = S   ∞                           = S   1
are both norms (the first is the maximal sum on a row of S ′ , and the second is the maximal sum on a
column of S ′ ), they verify
                                                    ′
                                           |λm | ≤ Sk .
                                                                                                                                ⊔
                                                                                                                                ⊓

    What it actually means is that we have two maximal1 sums: the one which can be computed on a row
of S ′ , and the one which can be computed on a column of S ′ . As these two maximums are norms, we have
to take the lowest value between them, to find the maximal bound for the eigenvalue of S ′ .
                      √   √
Remark 2. If S ′ is    n × n (n being a square number), we do not always have
                                                                                                                            
√                                                                          √        √                              √
                                                                                                                               
    n−1
                                       √                                        m= n                               n
                                                                                                                               
                                                                                               
          (n−i) = (n)+(n−1)+· · · +(n−( n−1)) =  max                                     ′
                                                                                         Si,j    or  max√
                                                                                                     j∈{1,··· , n}
                                                                                                                         ′
                                                                                                                        Si,j   ,
                                                                                                                               
                                                i∈{1,··· ,√n}                                                               
 i=0                                                                            j=1                               i=1

                                                                           = S   ∞                              = S   1


                                                                                         √                                     (7)
as there are valid Sudoku where at least one of the submatrices of size                      n where
                                                                        √
                                          (n) + (n − 1) + · · · + (n − ( n − 1))

equals neither S ′    1   nor S ′   ∞.                                                                                          ⊔
                                                                                                                                ⊓

Corollary 3 (Most general eigenvalue of a Sudoku submatrix). If S is the matrix of a n×n Sudoku,
with n being a square number (i.e. there exists one a ∈ N∗ so that a × a = n), the most general maximum
                                                   ′                                                   ′
(bounding) eigenvalue of one of its submatrices Sk , k ∈ {1, · · · , n}, verifies, if we consider that Sk = S ′ :
                                                                                       
                                     
                                                                                            
                                                                                             
                                     
                                                                                          
                 √
                                     
                                     
                                                     √
                                                               
                                                                                  √
                                                                                           
                   n−1
                                       
                                                       n
                                                          ′  
                                                                                    n
                                                                                        ′ 
                                                                                           
                       (n − i) ≥ min  max       √       Si,j  ,  max√              Si,j  ,                  (8)
                  i=0                i∈{1,··· , n} j=1
                                     
                                     
                                                               j∈{1,··· , n}
                                                                                 i=1      
                                                                                             
                                     
                                                                                            
                                                                                             
                                                                                            
                                                      = S   ∞                             = S    1



and                                                    √
                                                           n−1
                                                                 (n − i)                                                       (9)
                                                        i=0

is thus the most general maximum eigenvalue of a given Sudoku submatrix.                                                        ⊔
                                                                                                                                ⊓

Proof. Trivial. (Just take a counterexample.)                                                                                   ⊔
                                                                                                                                ⊓

Example 2 (9 × 9 Sudoku matrices). If we take a 9 × 9 Sudoku matrix, having 3 × 3 submatrices (which are
not Sudoku), the maximal sum one can encounter on a column or a row of a submatrix is
                                                                   √
                                         n + (n − 1) +        n − ( n − 1)             = 24.
                                         =9                       √
                                                 =8        =9−( 9−1)=9−(2)=7

                                                       ′
Consequently, the maximal eigenvalue one can find in a Sk , k ∈ {1, · · · , 9} is 24.                                            ⊔
                                                                                                                                ⊓
1
    For a given submatrix.
Example 3 (16 × 16 Sudoku matrices). Let’s take a 16 × 16 Sudoku matrix. Its submatrices are not Sudoku
matrices anymore, and are composed by elements in {1, · · · , 16} ⊂ Z. The maximal sum we can have, when
summing elements from one submatrix of a 16 × 16 Sudoku is

                                          16 + 15 + 14 + 13 = 58.
                                                            √
                                        four elements, as       16 = 4

                                                                                                      ⊔
It is the maximum real eigenvalue which can be found with the submatrices of a 16 × 16 Sudoku matrix. ⊓
Example 4 (25 × 25 Sudoku matrices). Let’s take a 25 × 25 Sudoku matrix. Its submatrices will never be
Sudoku matrices. They are composed by elements in {1, · · · , 25} ⊂ Z. The maximal sum we can have, when
summing elements from one submatrix of a 25 × 25 Sudoku is

                                      25 + 24 + 23 + 22 + 21 = 115.
                                                          √
                                       five elements, as       25 = 5

The maximum eigenvalue of a 5 × 5 (non-Sudoku) submatrix extracted from a 25 × 25 Sudoku matrix equals
thus 115.                                                                                           ⊔
                                                                                                    ⊓


6     Sudoku Eigenvectors
                                                                         n(n+1)
We can compute the corresponding eigenvector for the eigenvalue             2   ,   as computing

                                                n(n + 1)
                                           S−            I x = 0n
                                                   2
is the same as computing
                                                                           
                    S1,1 − n(n+1)
                              2        S1,2          ···          S1,n
                                                                           
                        S2,1     S2,2 − n(n+1)
                                            2        ···          S2,n      
                         .             .                          .         xn = 0n ,
                         .             .       ..    n(n+1)       .        
                         .             .          .− 2            .        
                                                                     n(n+1)
                         Sn,1          Sn,2          ···     Sn,n − 2

thus making computing
                                                                          
                  S1,1 − n(n+1)
                            2        S1,2           ···          S1,n         
                                       n(n+1)                              a
                      S2,1     S2,2 − 2            ···          S2,n       
                       .             .                           .          b  = 0n .           (10)
                       .             .        ..    n(n+1)       .         .
                       .             .           .− 2            .         .
                                                                    n(n+1)
                                                                               .
                       Sn,1          Sn,2           ···     Sn,n − 2


7     Other Facts About S
7.1   Transpose
General Facts One could ask if
                                                                                   
                                S1,1 S1,2    · · · S1,n     S1,1 S2,1      · · · Sn,1
                               S2,1 S2,2    · · · S2,n   S1,2 S2,2      · · · Sn,2 
                                                                                   
                            S= .     .      ..     . = .       .        ..     . 
                               ..    .
                                      .          . .   .
                                                    .        .    .
                                                                  .            . .. 
                               Sn,1 Sn,2     · · · Sn,n    S1,n S2,n       · · · Sn,n
is still a valid Sudoku matrix (i.e. it fulfills the rules of Sudoku).
    An example would be given by
                                                                        
                                        629385147               658421397
                                     5 1 3 7 6 4 8 2 9 2 1 4 9 7 3 8 6 5
                                                                        
                                     8 4 7 1 2 9 6 5 3 9 3 7 6 5 8 1 4 2
                                                                        
                                     4 9 6 2 3 1 7 8 5 3 7 1 2 9 6 4 5 8
                                                                        
                               B = 2 7 5 9 4 8 3 6 1 = 8 6 2 3 4 5 9 7 1
                                                                        
                                     1 3 8 6 5 7 4 9 2 5 4 9 1 8 7 2 3 6
                                                                        
                                     3 8 1 4 9 2 5 7 6 1 8 6 7 3 4 5 2 9
                                                                        
                                     9 6 4 5 7 3 2 1 8 4 2 5 8 6 9 7 1 3
                                        752816934               793512684

It is noticeable that B is still a valid Sudoku matrix, but B = B.
Theorem 7 (Transpose of a Sudoku matrix). The transpose of a Sudoku matrix is still a correct, but
different, Sudoku matrix (of the same dimension).                                                ⊔
                                                                                                ⊓
Proof. According to the rules of Sudoku, at least one number cannot be repeated at least two times on a
                          1      1
row, a column, or in a n 2 × n 2 square, where n is the dimension of the Sudoku. The matrix which is the
result of the transposition is still a valid Sudoku matrix, as
 1. Bi,j = Bi,j if i = j (i.e. (i, j) is a diagonal couple),
 2. If we let jold := j, j := i and i := jold , everywhere in B, we obtain B. By reversing rows and columns,
    rules of Sudoku are still respected, as the rules of Sudoku are applicable on both,
 3. For subsquares, rules are still respected, because of the last point.
We now have to prove that it is always different. We have to prove that there is no Sudoku matrix in S
which verifies Si,j = Si,j with 1 ≤ i ≤ n, and 1 ≤ j ≤ n. Let’s reason by contradiction. All the S matrices
verify
                                                Si,j = Si,j
for i = j. Let 1 ≤ i ≤ n, 1 ≤ j ≤ n. Is the same equality possible with these conventions? We shall now try
to construct such a matrix. Let’s reason first with a 4 × 4 matrix. We define
                                                                                         
                       B1,1 B1,2 B1,3 B1,4                              C1,1 C1,2 C1,3 C1,4
                     B B B B                                        C C C C 
                B :=  2,1 2,2 2,3 2,4 
                     B3,1 B3,2 B3,3 B3,4        and      B := C :=  2,1 2,2 2,3 2,4  .
                                                                      C3,1 C3,2 C3,3 C3,4 
                       B4,1 B4,2 B4,3 B4,4                              C4,1 C4,2 C4,3 C4,4

It is clear that Bi,j = Bi,j for all 1 ≤ i ≤ n and 1 ≤ j ≤ n iff Bi,j = Ci,j for all i and j. The C matrix has
thus to be exactly B. It is the case iff Bi,j = Bj,i , with 1 ≤ i ≤ n and 1 ≤ j ≤ n, thus iff B is symmetric.
Let’s try to construct a symmetric B. Such a B would be symmetric:
                                                                 
                                              B1,1 B2,1 B3,1 B4,1
                                            B2,1 B2,2 B3,2 B4,2 
                                                                 
                                            B3,1 B3,2 B3,3 B4,3  ,
                                              B4,1 B4,2 B4,3 B4,4
                                                                                                1     1
but it does not respect the rules of Sudoku, as there is at least one subsquare of dimension 4 2 × 4 2 ≡ 2 × 2
where two elements are the same (here, there is only one: B4,3 ). Here is the generalization of this fact. If we
take                                                               
                                               B1,1 B1,2 · · · B1,n
                                              B2,1 B2,2 · · · B2,n 
                                                                   
                                        B= .          . ..     . ,
                                              . .     .
                                                       .     . .. 
                                                Bn,1 Bn,2 · · · Bn,n
B should verify                                                        
                                              B1,1 B2,1      · · · Bn,1
                                             B2,1 B2,2      · · · Bn,2 
                                                                       
                                          B= .     .        ..     . 
                                             ..    .
                                                    .            . .. 
                                              Bn,1 Bn,2      · · · Bn,n
to be symmetric. It is symmetric, but does not respect the rules of Sudoku anmyore, as, at the place of the
.
. and the · · · , zooming at the end would lead to the submatrix
.
                                                                    
                                             ..
                                               . ··· ···      Sn,2 
                                           . ..
                                              .           .
                                                          .      . 
                                                                 . 
                                           .        .    .      . .
                                          
                                           .            ..          
                                           . ···
                                              .             . Sn,n−1 
                                            Sn,2 · · · Sn,n−1 Sn,n

                                                                                                           ⊔
                                                                                                           ⊓

Determinant
Theorem 8 (Determinant of the transpose of a square matrix). For every square matrix C,
det(C) = det(C).                                                                      ⊔
                                                                                      ⊓
This theorem is not proved here, even if it is evident.
Corollary 4 (Determinant of the transpose of a Sudoku matrix). If S is the matrix of a given
Sudoku, det(S) = det(S).                                                                   ⊔
                                                                                           ⊓

Corollary 5. If S is the matrix of a given Sudoku, S its transpose, and n their dimension (which is the
same),
                                     n(n + 1)                     n(n + 1)
                     det(S) mod                = det(S) mod                  = 0.
                                         2                            2
                                                                                                           ⊔
                                                                                                           ⊓

Proof. Trivial, as, on one hand, according to Theorem 7, S and S
 – have the same dimension (n),
                                   n              n(n+1)
 – are both Sudokus, verifying     i=1   Si,j =      2   ;
                                                                                  n(n+1)
on the other hand, Theorem 4 gives us the maximum eigenvalue of S, which is          2   .   Finally, we know
that
                                             λ=0
                                       det(S) = det(S − λI).
                                                                                                           ⊔
                                                                                                           ⊓

Trace
Theorem 9 (Trace of a transpose). Whatever the matrix E, Tr(E) = Tr(E).                                    ⊔
                                                                                                           ⊓
This theorem is not proved here, even if it is evident.
Corollary 6 (Trace of the transpose of a Sudoku matrix). If S is the matrix of a given Sudoku,
Tr(S) = Tr(S).                                                                              ⊔
                                                                                            ⊓
7.2   Not Hermitian
A matrix C is Hermitian if and only if
                                                                   C = C ∗.
As, for any Sudoku matrix S, S = S, as · is the complex conjugate of ·, and the Si,j , 1 ≤ i ≤ n, 1 ≤ j ≤ n,
asking if any Sudoku matrix is Hermitian or not is equivalent to ask if, for S, S = S. However, this is not
true, as we have
                                                  S = S,
thanks to Theorem 7.
Corollary 7 (Every Sudoku matrix is not Hermitian). There is no Hermitian Sudoku matrix.                                                             ⊔
                                                                                                                                                     ⊓

Proof. Direct using Theorem 7, as Si,j = Si,j for 1 ≤ i ≤ n, 1 ≤ j ≤ n, as Si,j ∈ R.                                                                 ⊔
                                                                                                                                                     ⊓
                                                                                                                                                     ⊔
                                                                                                                                                     ⊓

Corollary 8 (Every Sudoku matrix is not antisymmetric). S is not antisymmetric, for clear reasons:
Si,j = −Sj,i .                                                                                  ⊔
                                                                                                ⊓

Proof. The Si,j are in Z∗ . If we take −Si,j for 1 ≤ i ≤ n and 1 ≤ j ≤ n, all the Si,j will lie in Z∗ , or no
                                                                                                    −
Sudoku matrix can have negative elements.                                                                   ⊔
                                                                                                            ⊓

7.3   Not Normal
A matrix C is normal if and only if
                                                                CC ∗ = C ∗ C.
It is equivalent to ask, for now clear reasons,

                                                                 C C = CC,

if C is a Sudoku matrix.

Theorem 10 (Every Sudoku matrix is not normal). There is no normal Sudoku matrix.                                                                    ⊔
                                                                                                                                                     ⊓

Proof. Let’s try to build such a matrix. If we have
                                                                                                                          
                        S1,1 S1,2 · · · S1,n                                                S1,1 S2,1             · · · Sn,1
                       S2,1 S2,2 · · · S2,n                                              S1,2 S2,2             · · · Sn,2 
                                                                                                                          
                 S= .        . ..       .      and                           S ′ := S =  .     .               ..     . ,
                       . .   .
                              .       . . 
                                         .                                                 ..    .
                                                                                                  .                   . . 
                                                                                                                         .
                              Sn,1 Sn,2 · · · Sn,n                                          S1,n S2,n             · · · Sn,n

we have S S = SS iff
                                                                                                                                      
             S1,1 S1,2      · · · S1,n     S1,1 S2,1        · · · Sn,1     S1,1 S2,1             · · · Sn,1     S1,1 S1,2         · · · S1,n
            S2,1 S2,2      · · · S2,n   S1,2 S2,2        · · · Sn,2   S1,2 S2,2             · · · Sn,2   S2,1 S2,2         · · · S2,n 
                                                                                                                                      
      S= .        .        ..     .  .        .          ..     . = .       .               ..     .  .        .           ..     . ,
            ..    .
                   .            . ..  .   .    .
                                                 .              . ..   .  .    .
                                                                                 .                   . ..  .   .    .
                                                                                                                      .               . .. 
               Sn,1 Sn,2    · · · Sn,n    S1,n S2,n         · · · Sn,n    S1,n S2,n              · · · Sn,n    Sn,1 Sn,2          · · · Sn,n

thus asking an equality between
        0         2      2              2
                S1,1 + S1,2 + · · · + S1,n         S1,1 S2,1 + S1,2 S2,2 + · · · + S1,n S2,n   · · · S1,1 Sn,1 + S1,2 Sn,2 + · · · + S1,n Sn,n
                                                                                                                                                 1
        B
        BS                                                  2      2              2                                                            C
        B 2,1 S1,1 + S2,2 S1,2 + · · · + S2,n S1,n        S2,1 + S2,2 + · · · + S2,n           · · · S2,1 Sn,1 + S2,2 Sn,2 + · · · + S2,n Sn,n C
                                                                                                                                               C
        B                  ···                                       ···                       ···                     ···                     C
        @                                                                                                                                      A
         Sn,1 S1,1 + Sn,2 S1,2 + · · · + Sn,n S1,n Sn,1 S2,1 + Sn,2 S2,2 + · · · + Sn,n S2,n   ···           2       2              2
                                                                                                            Sn,1 + Sn,2 + · · · + Sn,n
and    0         2      2              2                                                                                                      1
               S1,1 + S1,2 + · · · + Sn,1         S1,1 S1,2 + S2,1 S2,2 + · · · + Sn,1 Sn,2   · · · S1,1 S1,n + S2,1 S2,n + · · · + Sn,1 Sn,n
                                                                  2
                                                        S1,2 2 + S2,2 + · · · + Sn,2 2
       B                                                                                                                                      C
       B 1,2 S1,1 + S2,2 S2,1 + · · · + Sn,2 Sn,1
       BS                                                                                     · · · S1,2 S1,n + S2,2 S2,n + · · · + Sn,2 Sn,n C
                                                                                                                                              C,
       B                  ···                                       ···                       ···                     ···                     C
       @                                                                                                                                      A
        S1,n S1,1 + S2,n S2,1 + · · · + Sn,n Sn,1 S1,n S1,2 + S2,n S2,2 + · · · + Sn,n Sn,2   ···           2
                                                                                                          Sn,1 + S2,n  2 + · · · + S2
                                                                                                                                    n,n

leading to such a system:
                                                              2            2
                                                             S1,n
                                                                        = Sn,1
                                                             
                                                             S
                                                              1,2
                                                                        = S2,1
                                                             
                                                              Sn,1       = S1,n .
                                                             
                                                             Sn,2
                                                             
                                                                        = S2,n
                                                             
                                                             
                                                                        .
                                                                         .
                                                                         .
It is clear that S has to be symmetric to respect these equalities. However, it was proven at Theorem 7 that
S cannot be symmetric.                                                                                    ⊔
                                                                                                          ⊓

7.4   Not Orthogonal
A matrix C is orthogonal if and only if
                                                                  CC = In .
Theorem 11 (Non-orthogonality of S if det(S) = 0). If det(S) = 0, S cannot be orthogonal.                                                          ⊔
                                                                                                                                                   ⊓
Proof. We have
                                                     det(S · S) = det(S) det(S),
which equals 0 if det(S) = 0, and det(In ) = 1.                                                                                                    ⊔
                                                                                                                                                   ⊓
If det(S) = 0, S has to coincide with S −1 to make S orthogonal. Using Theorem 7, S = S. But one might
ask the question “Can S equal S −1 ?” We know that S = S. Thus, if even S = S −1 , S −1 is different from S.
Theorem 12 (Non-orthogonality of S if det(S) = 0). If det(S) = 0, S cannot be orthogonal.                                                          ⊔
                                                                                                                                                   ⊓
Proof. The only way to make S orthogonal is to have S = S −1 . Or this equality is impossible: as

                                                   S −1 = cofact(S) · (det(A))−1 ,
it would be equivalent to ask
                                                     S = cofact(S) · (det(A))−1 ,
which is never true for a S Sudoku matrix.                                                                                                         ⊔
                                                                                                                                                   ⊓

8     Order
Theorem 13 (Every invertible matrix has a finite order). Every invertible matrix has a finite order.
                                                                                                ⊔
                                                                                                ⊓
This theorem is proved in many books, and its proof does not enter in the scope of this paper.
Corollary 9. If det(S) = 0, S has a finite order.                                                                                                   ⊔
                                                                                                                                                   ⊓
Example 5. The determinant
                                                                     1234
                                                                     3412
                                                            det(B) =
                                                                     4321
                                                                     2143
equalling 0, there is no smallest k > 0 such that B k = In .                                                                                       ⊔
                                                                                                                                                   ⊓
However, there is no special remark to do about this and Sudokus: there seems to be no link between the
order of a Sudoku matrix S and other concepts related to Sudoku’s matrices.
9   Trace
Let S be a (square) Sudoku matrix, and Si,i its diagonal elements (1 ≤ i ≤ n).
Theorem 14 (Upper bound on the trace of a Sudoku matrix). The value of Tr(S) verifies
                                                                n
                                                Tr(S) =              Si,i
                                                             i=1
                                                             √                       √n
                                                                     n−1
                                                            ≤              (n − i)        .
                                                                    i=1

                                                                                                               ⊔
                                                                                                               ⊓

Example 6 (9 × 9 Sudoku). For a 9 × 9 S Sudoku matrix, we have
                                           =2
                                                      3
                                          √
                                         9−1         
                           9                         
                                                                           3
                Tr(S) =         Si,i   ≤      (9 − i) = ((9)(9 − 1)(9 − 2)) = 128024064.
                                                     
                          i=1            i=0         


                                                                                                               ⊔
                                                                                                               ⊓

Remark 3 (Upper bound on the trace of a Sudoku matrix seen as a permutation). If Ak denotes the number
                                                                                    n
of combinations of k elements among n, the order of the elements being taken into account, we have
                                       √                    √n                                √
                                             n−1                                                    n
                                                                                 n!
                                                   (n − i)            =           √                    .
                                            i=1
                                                                              (n − n)!

It can be easily understood in another way than using pure Mathematics: we need to know how many
                                           √
different configurations exist to combine n − n elements among n, the order being taken into account.
This is achieved using
                                                 n!
                                                   √
                                             (n − n)!
              √
but there are n configurations which can lead to different results, at least if one of these configurations is
different from the others.                                                                                ⊔
                                                                                                         ⊓

Example 7 (Upper bound on the trace of a Sudoku matrix seen as a permutation for a 9 × 9 Sudoku matrix).
For a 9 × 9 Sudoku matrix, the upper bound on its trace is
                                        3               3                            3
                           9!                      9!               9 · 8 · 7 · 6!
                            √               =               =                            = 5043 = 128024064.
                       (9 − 9)!                    6!                     6!

                                                                                                               ⊔
                                                                                                               ⊓

Corollary 10 (Trace of a Sudoku matrix is similarity-invariant). The trace of a S Sudoku matrix
is similarity-invariant, i.e.
                                   Tr(S −1 AS) = Tr(A).
                                                                                                               ⊔
                                                                                                               ⊓
Corollary 11 (Trace of a Sudoku matrix equals the sum of the eigenvalues). We have
                                                            √                    √n
                                               n                     n−1
                                    Tr(S) =          λi ≤                 (n − i)     .
                                              i=1                 i=1

                                                                                                       ⊔
                                                                                                       ⊓

Corollary 12. We have
                                                    Tr(S ∗ S) ≥ 0.
                                                                                                       ⊔
                                                                                                       ⊓

Corollary 13. Let
                                                           k
                                              P (λ) =            (x − λi )di
                                                           i=1

be the characteristic polynomial of S. Then,
                                                               √                  √n
                                              n                      n−1
                                    Tr(S) =         di λ i ≤              (n − i)         .
                                              i=1                    i=1

                                                                                                       ⊔
                                                                                                       ⊓

Corollary 14 (Linear map). The trace being a linear map, if S and S ′ are Sudoku matrices, they verify

                                       Tr(S + S ′ ) = Tr(S) + Tr(S ′ )
                                          Tr(λS) = λ Tr(S), ∀ λ ∈ K.

Corollary 15. Let the biggest eigenvalue of a S Sudoku matrix be the jth eigenvalue, with j = n (that is,
the eigenvalues are classed with a ≤ order). Then,
                                                     n−1
                                                                        n(n + 1)
                                        Tr(S) =            di λ i +              .
                                                     i=1
                                                                           2

                                                                                                       ⊔
                                                                                                       ⊓

Proof. By definition,
                                                                 n
                                                  Tr(S) =             di λ i .
                                                               i=1

As the biggest eigenvalue of S is
                                                           n(n + 1)
                                                  λn =              ,
                                                              2
whose dn = 1 (both concepts according to Perron’s theorem (3) and to Theorem 4),
                                                     n−1
                                                                        n(n + 1)
                                        Tr(S) =            di λ i +              .
                                                     i=1
                                                                           2

                                                                                                       ⊔
                                                                                                       ⊓
9.1   Antitrace
Definition 1 (Antitrace of a square matrix). The antitrace of a square matrix C is
                                                               n
                                            antiTr(C) =            Ci,n−i+1 .
                                                           i=1

                                                                                                                  ⊔
                                                                                                                  ⊓
Theorem 15. Let S be a Sudoku matrix which the expression
                                                    √                   √n
                                                         n−1
                                           Tr(S) =            (n − i)
                                                         i=1

holds with. Then, S has its antitrace verifying
                                                         √                √n
                                                              n−1
                                         antiTr(S) <               (n − i)
                                                           i=1


and reciprocally for Tr(s) iff
                                                      √                   √n
                                                           n−1
                                        antiTr(S) =               (n − i)      .
                                                          i=1

                                                                                                                  ⊔
                                                                                                                  ⊓
Proof. Let’s try to build such a matrix. If Tr(S) is maximal, S, of the form
                                                                         
                                                     S1,1 S1,2 · · · S1,n
                                                  S2,1 S2,2 · · · S2,n 
                                                                         
                                           S= .           . ..       . ,
                                                  .  .    .
                                                           .       . . 
                                                                      .
                                                    Sn,1 Sn,2 · · · Sn,n
                      √                                               √
needs to have the n elements of the set {n, n − 1, · · · , n − n + 1} in each of its submatrices’ diagonals.
For example, a submatrix extracted from S must have the following form:
                                                                             
                                       Sp,q         Sp,q+1 · · · Sp,q+√n
                                    Sp+1,q Sp+1,q+1 · · · Sp+1,q+√n 
                                                                             
                                        .
                                         .             .
                                                       .      ..          .
                                                                          .   .
                                        .             .         .        .   
                                         √            √
                                     Sp+ n,q Sp+ n,q+1 · · · Sp+ n,q+ n √   √

                                                      √
If Tr(S) is maximal, the Sp+i,q+i , i ∈ {0, 1, · · · , n} must be the biggest numbers in {1, 2, · · · , n}, that is,
                     √
n, n − 1, · · · , n − n + 1. Consequently, there cannot be any other such numbers in this submatrix, as it
would result in an erroneous Sudoku, at least one of its submatrices having at least twice the same element.
The only way to reach
                                                √                  √n
                                                   n−1
                                                        (n − i)
                                                   i=1

is thus reserved to exactly one diagonal or no diagonal: the principal one (resulting in a maximal trace), the
other one (resulting in a maximal antitrace) or no one.                                                      ⊔
                                                                                                             ⊓
Example 8 (9 × 9 Sudoku matrix). A 9 × 9 Sudoku matrix has a maximal trace iff all its diagonal elements
are in {9, 8, 7}, each one being written only once in each of the diagonal submatrices. For example,
                                                                             
                                   S1,1 S1,2 S1,3 S1,4 S1,5 S1,6 S1,7 S1,8 9
                                 S2,1 S2,2 S2,3 S2,4 S2,5 S2,6 S2,7 8 S2,9 
                                                                             
                                 S3,1 S3,2 S3,3 S3,4 S3,5 S3,6 7 S3,8 S3,9 
                                                                             
                                 S4,1 S4,2 S4,3 S4,4 S4,5 9 S4,7 S4,8 S4,9 
                                                                             
                                 S5,1 S5,2 S5,3 S5,4 8 S5,6 S5,7 S5,8 S5,9 
                                                                             
                                 S6,1 S6,2 S6,3 7 S6,5 S6,6 S6,7 S6,8 S6,9 
                                                                             
                                 S7,1 S7,2 9 S7,4 S7,5 S7,6 S7,7 S7,8 S7,9 
                                                                             
                                 S8,1 8 S8,3 S8,4 S8,5 S8,6 S8,7 S8,8 S8,9 
                                    7 S9,2 S9,3 S9,4 S9,5 S9,6 S9,7 S9,8 S9,9

is a possible configuration, and
                                                                                                
                                    S1,1   S1,2   S1,3   S1,4   S1,5   S1,6   S1,7   S1,8    8
                                  S2,1    S2,2   S2,3   S2,4   S2,5   S2,6   S2,7    7     S2,9 
                                                                                                
                                  S3,1    S3,2   S3,3   S3,4   S3,5   S3,6    9     S3,8   S3,9 
                                                                                                
                                  S4,1    S4,2   S4,3   S4,4   S4,5    9     S4,7   S4,8   S4,9 
                                                                                                
                                  S5,1    S5,2   S5,3   S5,4    8     S5,6   S5,7   S5,8   S5,9 
                                                                                                
                                  S6,1    S6,2   S6,3    7     S6,5   S6,6   S6,7   S6,8   S6,9 
                                                                                                
                                  S7,1    S7,2    8     S7,4   S7,5   S7,6   S7,7   S7,8   S7,9 
                                                                                                
                                  S8,1     9     S8,3   S8,4   S8,5   S8,6   S8,7   S8,8   S8,9 
                                     7     S9,2   S9,3   S9,4   S9,5   S9,6   S9,7   S9,8   S9,9

is also a good one. They both lead to a maximal antitrace. However, the trace of S cannot equal the maximal
value
                                                 128024064
as, for example,                                                                                
                                    9      S1,2   S1,3   S1,4   S1,5   S1,6   S1,7   S1,8    8
                                  S2,1     8     S2,3   S2,4   S2,5   S2,6   S2,7    7     S2,9 
                                                                                                
                                  S3,1    S3,2    7     S3,4   S3,5   S3,6    9     S3,8   S3,9 
                                                                                                
                                  S4,1    S4,2   S4,3   9, 7   S4,5    9     S4,7   S4,8   S4,9 
                                                                                                
                                  S5,1    S5,2   S5,3   S5,4    8     S5,6   S5,7   S5,8   S5,9 
                                                                                                
                                  S6,1    S6,2   S6,3    7     S6,5   7, 9   S6,7   S6,8   S6,9 
                                                                                                
                                  S7,1    S7,2    8     S7,4   S7,5   S7,6    9     S7,8   S7,9 
                                                                                                
                                  S8,1     9     S8,3   S8,4   S8,5   S8,6   S8,7    8     S8,9 
                                    7      S9,2   S9,3   S9,4   S9,5   S9,6   S9,7   S9,8    7
is not a valid Sudoku matrix anymore.                                                                    ⊔
                                                                                                         ⊓
10     Condition Number
Theorem 16 (No well-conditioning for S can arise). The system Sx = b can be else than well-
conditioned (assuming b = 0n ).                                                          ⊔
                                                                                         ⊓

Proof. A definition of the condition number of S is

                         κ(S) = S −1     ∞   S   ∞
                                                 √                                           √                
                                                         n                                             n
                               =     max    √                   A−1  
                                                                  ij            max    √                   |Aij | .
                                   i∈{1,··· , n}                             i∈{1,··· , n}
                                                     j=1                                        j=1


1. If they are not equal, let’s call g the greatest, and s the smallest. We then have

                                              κ(S) = g ·                     s             .
                                                                       √
                                                                           n−1
                                                                   ≤             (n − i)
                                                                       i=0

     As g > s by hypothesis,
                                                                 κ(S) > 1
   can be true. As log(1) = 0, and that 1 > 0, a system of linear equations where the coefficient matrix is
   S will never be well-conditioned.
2. If they are equal, that is, if
                                                         √                                      √
                                                             n                                      n
                               p :=      max√                    A−1 =
                                                                  ij               max√                    |Aij | ,
                                      i∈{1,··· , n}                          i∈{1,··· , n}
                                                         j=1                                    j=1
                                               √                                           √             
                                                    n                                             n          
                        min  max                        A−1  ,              max                     |Aij | = p,
                            i∈{1,··· ,√n}                ij                          √
                                                                            i∈{1,··· , n}                     
                                               j=1                                             j=1

     we learn that
                                                 0 < κ(S) = p2 < +∞.
   That is, the system is never well-conditioned too (except iff p ∈ [0, 1[), but such a case does not arise
   often.
3. If S is not invertible, i.e. iff det S = 0, S −1 is infinite. Consequently, S −1 ∞ = ∞, and the system is
   thus ill-conditioned.
                                                                                                         ⊔
                                                                                                         ⊓

Example 9 (Extreme example). Consider the matrix
                                                           
                                            673514982
                                          9 2 1 3 6 8 7 5 4
                                                           
                                          5 8 4 7 9 2 1 3 6
                                                           
                                          8 9 6 2 3 7 4 1 5
                                                           
                                     P := 2 1 5 9 4 6 3 7 8 .
                                                           
                                          3 4 7 1 8 5 2 6 9
                                                           
                                          4 5 2 8 7 1 6 9 3
                                                           
                                          1 6 9 4 5 3 8 2 7
                                            738629541
It verifies

                                    κ(P )∞ ≈ (2.052475887800532) · 1017 ≫
                                    det(P ) = 0.

As κ(P ) ≫, trying to solve                               
                                       673514982           α       κ
                                      9 2 1 3 6 8 7 5 4 β  λ
                                                          
                                      5 8 4 7 9 2 1 3 6  γ  µ
                                                          
                                      8 9 6 2 3 7 4 1 5  δ   ν 
                                                          
                                      2 1 5 9 4 6 3 7 8  ǫ  =  ξ 
                                                          
                                      3 4 7 1 8 5 2 6 9  ζ  χ
                                                          
                                      4 5 2 8 7 1 6 9 3  η   ρ 
                                                          
                                      1 6 9 4 5 3 8 2 7  θ  σ 
                                       738629541            ι       τ
is really error-prone. Let’s take

                                                   κ=1
                                                   λ=2
                                                   µ=3
                                                   ν=4
                                                   ξ=5
                                                   χ=6
                                                   ρ=7
                                                   σ=8
                                                   τ = 9.

Using numerical analysis, we can now solve
                                                       
                                     673514982          α       1
                                   9 2 1 3 6 8 7 5 4 β  2
                                                       
                                   5 8 4 7 9 2 1 3 6  γ  3
                                                       
                                   8 9 6 2 3 7 4 1 5  δ  4
                                                       
                                   2 1 5 9 4 6 3 7 8  ǫ  = 5 .
                                                       
                                   3 4 7 1 8 5 2 6 9  ζ  6
                                                       
                                   4 5 2 8 7 1 6 9 3  η  7
                                                       
                                   1 6 9 4 5 3 8 2 7  θ  8
                                     738629541           ι      9

The result is roughly                                          
                                            −6.259323629462618
                                                               
                                            
                                                               
                                                                
                                             3.426654756202998 
                                                               
                                            
                                                               
                                                                
                                            −2.859392164516785
                                                               
                                            
                                                               
                                                                
                                            −0.213786330057330
                                         15
                                       10 · 2.235038905144813 .
                                             6.094125101691045 
                                                               
                                            
                                                               
                                                                
                                             0.146978101914415 
                                                               
                                            
                                                               
                                                                
                                            
                                                               
                                                                
                                             0.070452313314347 
                                                               
                                                               
                                             −2.640747054230881
Let’s now compute A := A + (0.1)9 and b := b + (0.1)9 . We now want to solve
                                                                 
                           6.1 7.1 3.1 5.1 1.1 4.1 9.1 8.1 2.1   α      1.1
                         9.1 2.1 1.1 3.1 6.1 8.1 7.1 5.1 4.1 β  2.1
                                                                 
                         5.1 8.1 4.1 7.1 9.1 2.1 1.1 3.1 6.1  γ  3.1
                                                                 
                         8.1 9.1 6.1 2.1 3.1 7.1 4.1 1.1 5.1  δ  4.1
                                                                 
                         2.1 1.1 5.1 9.1 4.1 6.1 3.1 7.1 8.1  ǫ  = 5.1 .
                                                                 
                         3.1 4.1 7.1 1.1 8.1 5.1 2.1 6.1 9.1  ζ  6.1
                                                                 
                         4.1 5.1 2.1 8.1 7.1 1.1 6.1 9.1 3.1  η  7.1
                                                                 
                         1.1 6.1 9.1 4.1 5.1 3.1 8.1 2.1 7.1  θ  8.1
                           7.1 3.1 8.1 6.1 2.1 9.1 5.1 4.1 1.1   ι      9.1

The result is now                                            
                                          −1.201790136856824
                                                             
                                           0.657917713190976 
                                          
                                                             
                                                              
                                                             
                                          −0.549003295587223
                                          
                                                             
                                                              
                                                             
                                          −0.041046975371007
                                          
                                                             
                                                              
                                    1017 · 0.429127469787804 .
                                           1.170072019524681 
                                          
                                                             
                                                              
                                                             
                                           0.028219795567568 
                                          
                                                             
                                                              
                                          
                                                             
                                           0.013526844156355 
                                                             
                                                              
                                          
                                                             
                                                              
                                           −0.507023434412329
                                                                                 ⊔
                                                                                 ⊓
11    Graph Modulo 2
Let’s say that Si,j := Si,j (mod 2), i.e. we now consider S modulo 2, for 1 ≤ i ≤ n, 1 ≤ j ≤ n. We now
consider S as the adjacency matrix of a graph G.
Theorem 17 (Possible diagonal of zeros). There are correct Sudoku matrices which, taken modulo 2,
have all their diagonal elements being equal to 0.                                             ⊔
                                                                                               ⊓

Proof. On one hand, consider the case of n = 4. It is easy to construct a Sudoku using digits 1, 2, 3 and 4
such that, modulo 2, all the diagonal elements equal 0:
                                                        
                                                 ab e f
                                                c d g h
                                                        
                                                i j m n ,
                                                 k l o p

where q means that q is even, when q means that q is odd. This matrix has, modulo 2, all its diagonal
elements which equal 0.
   On the other hand, take                                 
                                            593714286
                                          6 8 2 3 9 5 4 7 1
                                                           
                                          1 7 4 6 2 8 9 5 3
                                                           
                                          3 6 9 2 5 7 1 4 8
                                                           
                                     P := 8 1 7 4 6 9 3 2 5 .
                                                           
                                          4 2 5 8 3 1 7 6 9
                                                           
                                          9 4 1 5 8 2 6 3 7
                                                           
                                          2 3 8 9 7 6 5 1 4
                                            756143892
If we compute Pi,j (mod 2), 1 ≤ i ≤ 9, 1 ≤ j ≤ 9, we have
                                                            
                                             111110000
                                          0 0 0 1 1 1 0 1 1
                                                            
                                          1 1 0 0 0 0 1 1 1
                                                            
                                          1 0 1 0 1 1 1 0 0
                                                            
                                          0 1 1 0 0 1 1 0 1 .
                                                            
                                          0 0 1 0 1 1 1 0 1
                                                            
                                           1 0 1 1 0 0 0 1 1
                                                            
                                          0 1 0 1 1 0 1 1 0
                                             110101010
                                                                                                          ⊔
                                                                                                          ⊓

Theorem 18 (Possible symmetricity). There are correct Sudoku matrices which, taken modulo 2, are
symmetric. Sudoku matrices, taken modulo 2, of an even dimension, can be so, but Sudoku matrices, taken
modulo 2, of an odd dimension, are never symmetric.                                                   ⊔
                                                                                                      ⊓

Proof. On one hand, let’s take an example first. Consider the case of n = 4. It is easy to construct a Sudoku
using digits 1, 2, 3 and 4 such that, modulo 2, the derived adjacency matrix is symmetric: take for example
                                                          
                                                   ab e f
                                                  c d g h
                                                          
                                                  i j m n ,
                                                   k l o p
where q means that q is even, when q means that q is odd. This matrix verifies

                                      
                                      c
                                           (mod 2)    =b    (mod 2)
                                      
                                      
                                      i
                                           (mod 2)    =e    (mod 2)
                                      
                                      
                                      k     (mod 2)   =f    (mod 2)
                                                                                                      (11)
                                      
                                      j    (mod 2)    =g    (mod 2)
                                      
                                      
                                      l
                                           (mod 2)    =h    (mod 2)
                                      
                                      
                                      
                                       o    (mod 2)    =n    (mod 2)


   as it verifies
                                        
                                        (c − b)
                                                   (mod 2)    =0
                                        
                                        
                                        (i − e)
                                                   (mod 2)    =0
                                        
                                        
                                        (k − f )    (mod 2)   =0
                                        (j − g)
                                                   (mod 2)    =0
                                        
                                        
                                        (l − h)
                                                   (mod 2)    =0
                                        
                                        
                                        
                                         (o − n)     (mod 2)   =0


as the difference of two numbers is even (and thus equals 0 modulo 2) iff these two numbers are either
even, or odd. (This matrix was found under the hypothesis that o and n were either even, or odd, which
is true, as o = n if the matrix is symmetric.) This can be generalized to any even dimension, and such a
construction will always be possible because Sudoku matrices, taken modulo 2, of an even dimension, have
an even number of even numbers and an even number of odd numbers, whatever their submatrix.
    On the other hand, every Sudoku matrix taken modulo 2 of an odd dimension cannot be symmetric,
          √
as every n submatrix will have an odd number of odd numbers, and an even number of even numbers.
Consequently, there is a number, say x, which is a Si,j (mod 2), which make a difference of 0 modulo 2, i.e.
if x is the Sµ,θ and y the Sθ,µ ,

                                           (x − y) (mod 2) = 0,


as x and y cannot be simultaneously odd, and, consequently,


                                        x (mod 2) = y       (mod 2).


An example is given by
                                                             
                                             593714286
                                            6 8 2 3 9 5 4 7 1
                                                             
                                            1 7 4 6 2 8 9 5 3
                                                             
                                            3 6 9 2 5 7 1 4 8
                                                             
                                        P = 8 1 7 4 6 9 3 2 5 .
                                                             
                                            4 2 5 8 3 1 7 6 9
                                                             
                                            9 4 1 5 8 2 6 3 7
                                                             
                                            2 3 8 9 7 6 5 1 4
                                             756143892
If we compute Pi,j (mod 2), 1 ≤ i ≤ 9, 1 ≤ j ≤ 9, we have
                                                           
                                             111110000
                                          0 0 0 1 1 1 0 1 1
                                                           
                                          1 1 0 0 0 0 1 1 1
                                                           
                                          1 0 1 0 1 1 1 0 0
                                                           
                                          0 1 1 0 0 1 1 0 1 ,
                                                           
                                          0 0 1 0 1 1 1 0 1
                                                           
                                          1 0 1 1 0 0 0 1 1
                                                           
                                          0 1 0 1 1 0 1 1 0
                                             110101010

and it is not symmetric.                                                                                    ⊔
                                                                                                            ⊓

Corollary 16 (Graph of a Sudoku matrix can be simple or not). A graph which is generated using
the adjacency matrix modulo 2 of a Sudoku can be simple, or not.                             ⊔
                                                                                             ⊓

Proof. 1. It can be not simple: It can have at least one loop, as its diagonal can be filled with at least one
    “1;”
 2. It can be simple: It can have no multiple links and no loop, as its diagonal can be filled with zeros only.
                                                                                                            ⊔
                                                                                                            ⊓

Corollary 17 (Graph of a Sudoku matrix modulo 2 of an even dimension is not oriented). The
graph of a Sudoku matrix of an even dimension is never oriented.                         ⊔
                                                                                         ⊓

Corollary 18 (Graph of a Sudoku matrix modulo 2 of an odd dimension is always oriented).
The graph of a Sudoku matrix of an odd dimension is always oriented.                  ⊔
                                                                                      ⊓
References
1. Davis, Tom, The Mathematics of Sudoku, (2008). http://www.geometer.org/mathcircles/sudoku.pdf.
2. Frank, Richard, Mathematics in Sudoku, (2005). http://people.brandeis.edu/~kleinboc/47a/sudoku.pdf.
3. LeBoeuf, Robert, Properties of Sudoku and Sudoku Matrices, (2009). http://compmath.files.wordpress.
   com/2009/02/rlfreport.pdf.
4.      , Sudoku Matrices, (2009). http://compmath.files.wordpress.com/2009/02/rlposter.pdf.