HSC Science (Electronics) 11thMaharashtra State Board
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Concept: Symmetric and Skew Symmetric Matrices

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definition

1) - A square matrix A = `[a_(ij)]` is said to be symmetric if A′ = A, that is, `[a_(ij)]` = `[a_(ji)]` for all possible values of i and j. 
For example A = `[(sqrt3,2,3),(2,-1.5,-1),(3,-1,1)]`   is a symmetric matrix as A′ = A.

2) - A square matrix A = `[a_(ij)]` is said to be skew symmetric matrix if A′ = – A, that is `a_(ji)` = – `a_(ij)` for all possible values of i  and j. Now, if we put i = j, we have `a_(ii)` = `– a_(ii)`. Therefore `2a_(ii)` = 0 or `a_(ii)` = 0 for all i’s.
This means that all the diagonal elements of a skew symmetric matrix are zero.

theorem

For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew symmetric matrix.  

Proof:  Let B = A + A′, then
B′ = (A + A′)′ 
= A′ + (A′)′ (as (A + B)′ = A′ + B′) 
= A′ + A (as (A′)′ = A) 
= A + A′ (as A + B = B + A)
= B 
Therefore B = A + A′ is a symmetric matrix 
Now let C = A – A′ 
C′ = (A – A′)′ = A′ – (A′)′     (Why?) 
= A′ – A    (Why?) 
= – (A – A′) = – C
Therefore C = A – A′ is a skew symmetric matrix.

theorem

Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
Proof:  Let A be a square matrix, then we can write
A = `1/2` (A + A') + `1/2` (A - A')
we know that (A + A′) is a symmetric matrix and (A – A′) is
a skew symmetric matrix. Since for any matrix A, (kA)′ = kA′, it follows that `1/2`(A + A')is symmetric matrix `1/2`(A - A') is skew symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
Video link : https://youtu.be/v5abfTlztTc

Shaalaa.com | Matrices part 29 (Symmetric matrices)

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Matrices part 29 (Symmetric matrices) [00:04:09]
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