# Corporate Profile by aP68V1sd

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```									Matrix Algebra and Regression

2   5   6          matrix element
A   =
a13 = 6
1   2   3

•   a matrix is a rectangular array of elements
•   m=#rows, n=#columns  m x n
•   a single value is called a ‘scalar’
•   a single row is called a ‘row vector’ B = 12 25 91 30
•   a single column is called a ‘column vector’
Matrix Algebra and Regression

•    a square matrix has equal numbers of rows and columns
•    in a symmetric matrix, aij = aji       2    0   4   0   11
0    6   0   0   0
4    0   3   8   9
0    0   8   1   0
11   0   9   0   8

•    in a diagonal matrix, all off-diagonal elements = 0
•    an identity matrix is a diagonal matrix with diagonals = 1
1   0   0   0

I=       0   1   0   0
0   0   1   0
0   0   0   1
Trace

• The trace of a matrix is the sum of the
elements on the main diagonal
2   0   4   0   11
0   6   0   0   0
A=      4   0   3   8   9
0   0   8   1   0
11 0    9   0   8

tr(A) = 2 + 6 + 3 + 1 + 8 = 20

4   6   2       9   2   5       13   8    7
+               =
8   3   0       4   7   1       12 10     1

4   6   2       9   2   5       -5   4   -3
-               =
8   3   0       4   7   1       4    -4 -1

•   The dimensions of the matrices must be the same
Matrix multiplication

A                       B                   C
mxn                     nxm                 mxm
2   5   1     8         2    3    7            94   77   72
3   6   9     4         5    6    3            77 142 86
X                   =
7   3   3     5         1    9    3            72   86   92
8    4    5

C11 = 2*2 + 5*5 + 1*1 + 8*8 = 94
Matrix operations

•   To transpose a matrix, exchange rows and columns

2   5   6                        2      1
A   =                      A'    =       5      2
1   2   3                        6      3

•   the inverse of a matrix is analagous to division in
math
6   0   0                1/6 0      0
A   =   0   3   0      A-1   =   0 1/3 0
0   0   9                0       0 1/9
Inverting a 2x2 matrix

a     b                  2     5
M=                   M=
c     d                  3     9

D = ad - bc             D = 2*9 – 5*3

M-1 =   d/D    -b/D               9/3   -5/3
M-1   =
-c/D   a/D               -3/3   2/3
Linear dependence

a     b                         2     6
M=                             M=
c     d                         3     9

D = ad - bc                 D = 2*9 – 6*3 = 0

The matrix is singular because one row (or column) can be
obtained by multiplying another by a constant
The rank of a matrix = the number of linearly independent
rows or columns (1 in this case)
A nonsingular matrix is full rank and has a unique inverse
A generalized inverse (M–) can be obtained for any matrix
MM–M = M
Regression in matrix notation

Linear model          Y = X + ε

Parameter estimates   b = (X’X)-1X’Y

Source         df     SS               MS

Regression     p      b’X’Y            MSR

Residual       n-p    Y’Y - b’X’Y      MSE

Total          n      Y’Y

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