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# Symmetric and Skew Symmetric Matrices

#### 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.

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