Writing a Matrix as sum of Symmetric & Skew Symmetric matrix

For any square matrix A,

  (A + A’) is a symmetric matrix

  (A − A’) is a skew-symmetric matrix

Let’s first prove them

(A + A’) is a symmetric matrix

For a symmetric matrix

  X’ = X

So, we have to prove

   (A + A’)’ = (A + A’)

Solving LHS

Therefore,

  (A + A’)’ = A + A’

So, A + A’ is a symmetric matrix

(A − A’) is a symmetric matrix

For a skew symmetric matrix X

  X’ = −X

So, we have prove

      (A − A’)’ = − (A − A’)

Solving LHS

Therefore,

  (A − A’)’ = − (A − A’)

So, A − A’ is a skew symmetric matrix

Now,

Let’s write matrix A as sum of symmetric & skew symmetric matrix

       (A + A’) + (A − A’) = 2A

So,

  1/2 [(A + A’) + (A − A’)] = A

  1/2 (A + A’) + 1/2 (A − A’) = A

Here,

      1/2 (A + A’) is the symmetric matrix

&   1/2 (A − A’) is the symmetric matrix

Let’s take an example,

Let’s check if they are symmetric & skew-symmetric