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Symmetric and Skew Symmetric Matrices

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Symmetric and Skew Symmetric Matrices - Lesson Summary

Transpose of a matrix :

If A = [aij] be an m x n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A.

Ex : A = 2 1 -4 3 2 5   B =  2 3 1 2 -4 5            
       
The transpose of a matrix, A, is denoted as A' or AT.

(i) (X' )' = X,

(ii) (kX' )' = kX (where kis any constant)

(iii) (X + Y)' = X' + Y'

(iv) (X Y)' = Y' X'


Let X =  1 1 2 3 1 5      Y =  2 1 5 1 4 2  are  two matrices

i) (A')' = A

X =  1 1 2 3 1 5  

X' = 1 3 1 1 2 5

(X')' =  1 1 2 3 1 5   = X
        
(ii) (kA')' = kA (where kis any constant)

LHS: (5Y)'

5 x Y = 5 x  2 1 5 1 4 2  = 10 5 25 5 20 10   

(5Y)' =  10 5 5 20 25 10     

RHS: 5Y'

Y' = 2 1 1 4 5 2

5 x Y' = 5 x  2 1 1 4 5 2   =   10 5 5 20 25 10 


(5Y)' = 10 5 5 20 25 10     = 5Y'   

LHS = RHS

(iii) (A + B)' = A' + B'

(X + Y)'

X + Y =   1 1 2 3 1 5  + 2 1 5 1 4 2     =  3 2 7 4 5 7     

(X+Y)' =  3 4 2 5 7 7     

X' + Y' =  1 3 1 1 2 5   +  2 1 1 4 5 2     = 3 4 2 5 7 7  

(X+Y)' =  3 4 2 5 7 7    = X' + Y'

(iv) (AB)' = B' A'

X =  0 1 2 and Y = 1 5 7

LHS: (XY)'

XY =  0 1 2  1 5 7   =   0 0 0 1 5 7 2 10 14      

(XY)' = 0 1 2 0 5 10 0 7 14

RHS: Y' X'

X =   0 1 2   X' =  0 1 2 

 Y =   1 5 7   Y' =  1 5 7  

Y'X' = 1 5 7    0 1 2  = 0 1 2 0 5 10 0 7 14           

(XY)' =    0 1 2 0 5 10 0 7 14      = Y'X'

Symmetric matrix :

A square matrix A = [aij] is said to be symmetric if A' = A, that is, [aij] = [aij] for all possible values of i and j.

Ex: X = -1 2 5 3 4 6 5 6 1


0 1 2 0 5 10 0 7 14        Transpose operation →           0 1 2 0 5 10 0 7 14  

X' = X , X is a symmetric matrix.


Skew symmetric matrix:

A square matrix A = [aij] is said to be skew symmetric if A' = - A that is, [aij] = - [aij] for all possible values of i and j.

Ex: Y =  0 -2 5 2 0 -1 -5 1 0    

     0 -2 5 2 0 -1 -5 1 0           Transpose operation →              0 2 5 -2 0 1 -5 -1 0                                     

Y' = -Y ⇒ Y is a skew symmetric matrix.


Theorem 1: For any square matrix A with real number entries, A + A′ is a symmetric matrix and A - A′ is a skew symmetric matrix.

Proof1): A is a square matrix with real numbers as its elements.

Given, A = [aij]mxn

Let X = A + A'

If X is a symmetric matrix,

X = X'

X' = (A + A')'

= (A)' + (A')'  '.' [(A + B)' = A' + B']

= A' + A

= A' + A = X

X is a symmetric matrix or (A' + A) is a symmetric matrix

Proof 2): A - A' is a skew symmetric matrix.

Let Y = A - A'

Skew symmetric matrix: Y' = - Y

Y' = (A - A')'

= (A)' - (A')'

= A' - A

= - (A - A') = -Y

Y is a skew symmetric matrix or (A - A') is a skew symmetric matrix.

Ex: A =  2 3 5 4 3 2 1 0 7        A' =  2 4 1 3 3 0 5 2 7

To verify that A + A' is symmetric.

A + A' =  4 7 6 7 6 2 6 2 14       

  (A + A' ) ' =  4 7 6 7 6 2 6 2 14      Transpose operation →      4 7 6 7 6 2 6 2 14   = A + A'   

To verify that A - A' is a skew symmetric matrix.

A - A' = 0 -1 4 1 0 2 -4 -2 0            

0 -1 4 1 0 2 -4 -2 0          Transpose operation →        0 1 4 -1 0 -2 -4 2 0           

0 1 4 -1 0 -2 -4 2 0   = (-1) 0 -1 4 1 0 2 -4 -2 0  = -(A - A' )      

Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Proof: Let A = [aij]mxm

We know that A + A' : Symmetric matrix

                      A - A' : Skew Symmetric matrix

= ½(A + A' ) + ½ (A - A')

= (½A + ½A') + (½A - ½A') Using kA = k(A)

= ½A + ½A = A.

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