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					    Unit Two
B.Tech Ist Semester
Content
 Elementary Transformations (Operation)
 Rank of The matrix
 Normal Form
 Solution of System of Linear Eqn.
 Vectors
 Linear dependence & independence
 Linear &Orthogonal transformation
Content
 Eigen value, Eigen vectors and there properties
 Cayley-Hamilton Theorem
 Diagonal Form of Matrix
 Similar Matrix
 Quadratic form of Matrix
Matrix             A matrix is a rectangular array of numbers. We
                  subscript entries to tell their location in the array
aij " i th row"
" j th column"
                                 rows             m× n
   row    a11 a12 a13                 K  a1n 
         a    a22 a23                        
                                       K a2 n 
           21

     A =  a31 a32 a33                 K a3n 
                                                          Matrices are

          M    M                        M                 identified by
                                                              their size.
         am1 am 2 am 3
                                      K amn 
Matrix
  1× 5 [3 − 1 5 0 2]
                             Row Matrix

   4× 1                               4× 4
     − 2                       2 − 1 − 2 4
    6                          −1 3       
             Column Matrix




                                         5 7
                              
    1                         − 2 5 − 8 9 
                                          
                                4   7   9 0
     − 3                           Square Matrix
Matrix
 A matrix that has the same number of rows as
 columns is called a square matrix.



             a11   a12   a13   a14 
            a      a22   a23       
                                a24 
         A=  21
             a31   a32   a33   a34 
                                   
            a41    a42   a43   a44 
Matrix
   Diagonal Matrix                                          Unit Matrix
   a11 0 0 ............0                                1 0 0 ............0 
   0 a 0 ............0                                  0 1 0 ............0 
         22                                                                        
   0 0 a33 ...........0                                 0 0 1............0 
                                                                                   
   ...........................                          ...........................
   ...........................                          ...........................
                                                                                   
   
   0 0 0                0 ann  n×n
                                
                                                          
                                                          0 0 0 ........ 0 1  n×n   
                                   0 0 0 ............0 
                                   0 0 0 ............0 
                                                              
                                   0 0 0............0 
                                                               Null Matrix
                                   ...........................
                                   ...........................
                                                              
                                   0 0 0 ........ 0 0  n×n
                                                              
Determinant
A determinant is a real number associated with every
 square matrix. The determinant of a square matrix A is
 denoted by "det A" or | A |.
                                 a b
The determinant of 2x2 matrix is         = ad − bc
                                    c      d
and 3x3 matrix is
        a1 a2 a3
                        b2 b3           b3 b1          b1 b2
        b1 b2 b3 = a1            − a2           + a3
                        c 2 c2          c3 c1          c1 c2
        c1 c2 c3
e.g:
                                        1 2 1
                                       − 1 3 4
(a) Find the determinant of the matrix        .
                                        5 2 1
                                              
      a a2      a3 −1
(b) If b   b2   b 3 − 1 = 0 in which a,b,c are different,
       c   c2   c3 − 1

show that abc = 1 .
Properties of the determinant:
i.  A determinant of a matrix is unaffected by changing
    its rows(columns) into columns (rows).
ii. If two rows(columns) of a determinant are
    interchange then the numerical value of determinant
    remains same but change its sign.
iii. If two rows(columns) of a determinant are identical then the
    value of determinant is zero.
iv. If each element of any row(column) are multiplied by the same
    factor ,then whole determinant multiplied by same factor.
v. If each element of a column consist m terms ,the determinant
    can be expressed as the sum of m determinants.
Elementary Transformations
   Operations that can be performed without
   altering the solution set of a linear system


1. Interchange any two rows
2. Multiply every element in a row by a nonzero constant
3. Add elements of one row to corresponding
elements of another row
Some Result of E-operation
i.    Any E-row operation on the product of two matrices
      is equivalent to the same operation on the pre-factor.
      i.e. for E-row operation R, R(AB)=R(A).B .


ii.   Any E-column operation on the product of two
      matrices is equivalent to the same operation on the
      post-factor.
      i.e. for E-column operation C, C(AB)=A.C(B).
Inverse of a Matrix
 If A is a square matrix s.t. |A|≠0, then a matrix B is
 called the inverse of the matrix A if
                     AB=BA=I (identity matrix)
  It is denoted by A-1 .
Inverse of a matrix by E-operation
(Gauss-Jordan method):
 Let A be a square matrix s.t. |A|≠0, then A=IA
Appling E-row operation on A=IA, to reduce L.H.S.
  matrix A in to Identity matrix and in R.H.S. Identity
  matrix I reduced in a new matrix B(say).i.e.
         R(A)=R(I).A reduced to I=BA…………………(i)
   Post multiplying (i) by A-1 , we have

                   A-1 =B
e.g. :
  Use Gauss-Jordan method to find the inverse of the
         2 0 − 1
  matrix 5 1 0  .
                
         0 1 3 
                
Echelon Form
  All nonzero rows (rows with at least one nonzero
  element) are above any rows of all zeroes, and
  The leading coefficient (the first nonzero number
  from the left, also called the pivot) of a nonzero row is
  1 and always strictly to the right of the leading
  coefficient of the row above it.
   All entries in the column below a leading 1 are zero.

                 1      #     #      #
                 0      1     #       
                                      #
                 
                 0
                        0     1      #
                                       
e.g.: To change the Matrix A in echelon form by elementary
      transformation.
                          5 3 14 4
                          0 1 2 1 
                        A=         
                          1 − 1 2 0
                                   
Minor of Matrix
 If A be mxn matrix then determinant of sub matrix of a
 matrix A is called the Minor of the matrix.



e.g.: Find all the minors of the matrix
              2 1 − 3 − 6
             3 − 3 1   2
                         
             1 1
                   1   2 .
                          
Rank of the Matrix
 The order of the highest non-zero minor of a mxn
 matrix is called Rank of the matrix and is denoted by
 ρ(A).



Note: Number of non-zero row in the echelon form of
 the matrix is also the rank of matrix.
e.g.:
 (a) Determine the rank of the following matrix:
     0   1   −3   − 1
     1   0   1    1
                     
     3   1   0    1
                     
     1   1   −2   0

(b) Reduce the following matrix in Normal Form:
  2 1 − 3 − 6
  3 − 3 1  2
             
  1 1
        1  2
Normal form of a Matrix
If A is an mxn matrix and by a series of E-operation, it
  can be find in the following form

 I r   0 I r 
 0       , 0  , [I r 0], [I r ] ,where Ir is the unit
        0  
 
matrix of order r.
Note:
 For mxn matrix A of rank r, to find square matrices P &
 Q of order m and n respectively, such that PAQ is in
 normal form.
  P and Q can be formed by applying E-operation on
 A=IAI.
                           1 1 2 
e.g.: For the matrix   A = 1 2 3 
                                       find the non-
                           0 − 1 − 1
                                    
  singular matrix P and Q such that PAQ is in normal
  form.
Solution of a system of linear
Equation
                                     a1 x + b1 y + c1 x = d1 
Consider of a system of linear eqn . a2 x + b2 y + c2 x = d 2 
                                                              
In matrix notation we have           a3 x + b3 y + c3 x = d 3 
                                                              
          AX = B …………..(i)
              a1 b1 c1 
Where A = a2 b2 c2  is called the co-efficient
                        
 matrix.      a3 b3 c3 
                                    d1 
     x
                                     d  is determine
X =  y  is variable matrix and B =  2 
      matrix.
    z                              d 2 
                                      
     
If B≠0 then eqn. (i) is called non-homogeneous system
of equation, if B=0 then (i) is homogeneous system of
equation.
Solution of non-homogeneous system of linear eqn.
For a system of non-homogeneous linear equation AX=B
i.    If ρ[A:B]≠ρ(A), the eqn. have no solution, system is
     called inconsistent.
ii. If ρ[A:B]=ρ(A)= number of variable, the system has a
     unique solution.
iii. If ρ[A:B]=ρ(A)< number of variables, the system has
     an infinite no. of solutions.
Solution of homogeneous system of linear eqn.
For a system of homogeneous linear equations AX=0
   X=0 is always a solution, is called trivial solution.
 Thus a homogeneous system is always consistent.
 A system of homogeneous linear equation has either
 the trivial solution or infinite number of solutions.
 (i) if ρ(A)= number of variable, the system has only
 trivial solution.
  (ii) if ρ(A)< number of variable, the system has an
 infinite number of non-trivial solution.
e.g:
a) Test the consistency of the equation x+2y-z=3,3x-
       y+2z=1, 2x-3y+7z=5, 2x-2y+3z=2, x-y+z=-1, and
       solve them.


b) For what values of λ the equation 3x-y+4z=3, x+2y-
       3z=-2, 6x+5y+ λz=-3 have
       (i) no solution.
       (ii) a unique solution
       (iii) an infinite number of solution.
Vector
  Any ordered n-tuple of numbers is called an n-vector.
 If x1 , x2 , ................xn be any n numbers then the ordred
 n-tuple X = ( x1 , x2 ,...............xn ) in called n-vector.




 e.g.: In 3D the coordinates are represent with 3-
 vectors (x,y,z).
Linearly dependent and independent of
vector.:
A set of r vectors X1, X2, …………., Xr is said to be linearly
  dependent if there exist r scalars k1, k2, ………., kr not all
  zero , such that
               k1 X 1 + k 2 X 2 + ......... + k r X r = 0
 and X1, X2, …………., Xr is said to be linearly
 independent, ∀     k1 X 1 + k 2 X 2 + ......... + k r X r = 0
                  ⇒       k1 = k 2 = ......... = k r = 0
Note: Let X1, X2, …………., Xr are linearly dependent
vectors, such that, for r scalars k1, k2, ………., kr not all
zero k1 X 1 + k 2 X 2 + ......... + k r X r = 0
if ki ≠ 0 , then

    ki X i = − k1 X 1 − k 2 X 2 − ..... − ki −1 X i −1 − ki +1 X i +1....... − k r X r
             k1    k2                ki −1          ki +1             kr
    ⇒ X i = − X 1 − X 2 − ........ −       X i −1 −       X i +1...... X r
             ki    ki                 ki             ki               ki
then Xi is a linear combination of the remaining
vectors k1, k2, ……… ki-1, ki+1 ,………..kr .
Linear Transformation
If T : R n → R m is a linear transformation, then there
  exist a unique mxn matrix A, s.t.
           T(X)=AX for all      X ∈ Rn .
Let P(x,y) in a plane transformation, transform to the
  point P’(x’,y’) under the reflection in co-ordinate axes,
  then the co-ordinate of P’ can be expressed in terms of
  those of P by the linear transformation of the form
x' = a1 x + b1 y
                   or in matrix notation  x' =  a1 b1   x 
                                          y ' a b   y 
 y ' = a2 x + b2 y
                     or X’=AX .             2 2  
Orthogonal Transform
The linear transformation Y=AX, where
     y1      a11 a12 ..........          a1n          x1 
    y       a                                         x 
     : , A =  21 a22 ..........
       2                                   a2 n , X =  :2 
 Y=
    :       ...... ....... ...........   .....       :
                                                      
     yn 
             am1 am 2 ...........        amn          xn 
                                                          
is said to be orthogonal if it transformation

   2       2                    2                2        2           2
 y1 + y2 + ........... + yn         in to x1 + x2 + .............. + xn .
Note : If the transformation matrix is non-singular , i.e.
  if |A|≠0, then the linear transformation is called non-
  singular or regular transformation, if A is singular, i.e.
  |A|=0 then is called singular.

  e.g.: Show that the transformations
                y1 = x1 − x2 + x3
                 y2 = 3 x1 − x2 + 2 x3
                 y3 = 2 x1 − 2 x2 + 3 x3

   is non-singular. Find the inverse transformation.
Eigen value
Consider a linear transformation Y=AX. If Y be a matrix
 of some scalar multiple(say λ) of X, i.e. Y= λX ,
  then AX = λX ⇒ AX − λIX = 0
           ⇒ ( A − λI )X = 0...................................(i )
   eqn. (i) is system of homogeneous linear eqn. These
 system will have non-trivial solution if the coefficient
 matrix is singular, i.e. | A − λI |= 0...............(ii )
 eqn. (ii) are called Characteristic Equation. and
 having n roots ,say λ1, λ2,……….., λn which are called
 Eigen value or Characteristic Roots or Latent
 Roots.
Eigen vectors
For every value of λ, the homogeneous system of eqn.(i)
  have non-zero solution
      a1 
      a 
       2
      . 
   X =       which is called an Eigen vector corresponding
      . 
      .      to that value of λ.
       
       an 
       
e.g.: Find the Eigen value and Eigen vector of the
  matrix.
              − 2 2 − 3
             2    1 − 6
(i)                    
              −1 − 2 0 
                       



              8 −6 2 
(ii)          − 6 7 − 4
                       
              2 −1 3 
                       
Properties of Eigen Value.
(i) The eigen value of square matrix A and its transpose A’
        are the same.
(ii)    The sum of the eigen values of a matrix is the sum of the
        elements on principal diagonals.
(iii)   The product of the eigen values of a matrix A is equal to
        |A|.                                                             1
(iv)    If λ is an eigen value of a non-singular matrix A, then is
        an eigen value of of A-1 .                                       λ
(v)     If λ is an eigen value of an orthogonal matrix A, then is 1
        an eigen value of of A-1 .                                        λ
(vi)     If λ1 , λ2 ,........, λn are the eigenmvalues of a matrix A, then
        A m has the eigen values λ m , λ ,........, λ. m
                                          1   2          n  where m∈ Z +
Cayley Hamilton Theorem
Every square matrix satisfies it’s characteristic equation.
i.e. if the characteristic equation of the nth order matrix A
   is , | A − λI |= (−1) n λn + a1λn −1 + a2 λn − 2 + ............. + an = 0
 then (−1) n An + a1 An −1 + a2 An −2 + ............. + an I = 0


Note: According to Cayley Hamilton theorem
        (−1) n An + a1 An −1 + a2 An − 2 + ............. + an I = 0
Multiplying this by A-1 we have
(−1) n An −1 + a1 An − 2 + a2 An −3 + ............. + an −1 I + an A−1 = 0
               1
⇒ A = − [(−1) n An −1 + a1 An − 2 + a2 An −3 + ............. + an −1 I ]
       −1

              an
e.g.: (a)Verify Cayley Hamilton theorem for the matrix A
  and find A-1   7   2 − 2
       where A = − 6 − 1 2  .
                           
                 6
                     2 − 1
                                                       1 4 
(b) Find the characteristic equation of the matrix A = 
                                                        2 3
                                                            
 and use it to find the matrix represented by
  A5 + 5 A4 − 6 A3 + 2 A2 − 4 A + 7 I
Also express A5 − 4 A4 − 7 A3 + 11A2 − A − 10 I as a linear
  polynomial in A.
Diagonalization of Matrix
If a square matrix A of order n has n linearly independent
   eigen vectors then there exist a matrix B such that B-
   1AB is a diagonal matrix.



                                 8 −6 2 
                            A = − 6 7 − 4
e.g.: Diagonalise the matrix             
                                 2 −4 3 
                                         
Similar Matrix
 Let A and B be square matrices of the same order. The
 matrix A is said to be similar to the matrix B if there
 exist an invertible square matrix P, s.t. A=P-1BP.



e.g.: Show that the matrices
          5 5           1        2
      A=       and B =             are similar.
          − 2 0              − 3 4
Quadratic Form
A real quadratic form is an homogeneous expression of
 the form                n   n
                 Q = ∑∑ aij xi x j in which the total
                       i =1 j =1
                                 power in each term is 2.
For instance, if     a h f               x
                 A =  h b g  , X =  y  and X ' = [x y z ]
                                         
                       f
                            g   c
                                          z
                                            
then X ' AX = ax 2 + by 2 + cz 2 + 2hxy + 2 gxz + 2 fyz
which is quadratic form.
Canonical Form
                                       n    n
If the quadratic form X ' AX = ∑∑ aij xi x j can be reduced to
                                      i =1 j =1

                              n
the quadratic form Y ' BY = ∑ λi yi 2 by a non-singular linear
                            i =1
transformation X=PY then Y’BY is called the canonical
form of given one.
If B = P' AP = diag (λ1 , λ2 ,............, λn ) , then
                        n
    X ' AX = Y ' BY = ∑ λi yi
                                  2

                       i =1
                                  . Where B=P’AP.
Index and Signature of the
Quadratic form
The number of positive terms in the canonical form is
 called the index of the quadratic form.
Let p be the number of positive terms and rank of
 matrix is r, then the signature of matrix =2p-r.



e.g.: Reduce3x 2 + 5 y 2 + 3z 2 − 2 xy − 2 yz + 2 zx quadratic
  forms to canonical forms by orthogonal
  transformation. Write also rank, index and signature.
Nature of Quadratic form
A real quadratic form X’AX in n variables is said to be
i. positive definite if all the eigen values of A is >0.
ii. negative definite if all the eigen values of A is <0.
iii. positive semi-definite if all the eigen values of A is
     ≥0 and at least one eigen value =0.
iv. negative semi-definite if all the eigen values of A is
     ≤0 and at least one eigen value =0.

				
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posted:10/27/2012
language:English
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