Symmetric Matrix
A square matrix A = [aij] is said to be symmetric if A’ = A, that is, [aij] = [aji] for all possible values of i and j
For example:
Since A = A’, A is symmetric.
Skew - symmetric Matrix
A square matrix A = [aij] is said to be skew symmetric if A’ = -A, that is aji = -aij for all possible values of i and j.
For example:
Here B = B’, so it is skew symmetric.
Important Results
1) For any square matrix A with real number entries, A + A’ is a symmetric matrix and A - A’ is a skew symmetric matrix.
Example: For the matrix
(i) (A + A’) is symmetric matrix
(ii) (A - A’) is skew symmetric matrix
Solution:
From (a) and (b),
(A + A’) = (A + A’)’, so it is symmetric matrix
From (c) and (d),
(A - A’) = -(A - A’)’, so it is skew symmetric matrix.
2) Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.
Example: Express
As the sum of symmetric and a skew- symmetric matrices
Solution:
Let P = ½ [A + A’] and Q = ½ [A - A’]. We have to show that A = P + Q
Hence P + Q = A, so A is expressed as sum of a symmetric and skew symmetric matrices.
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