Matrix as a Sum of Symmetric Matrix and Skew-Symmetric Matrix

Every square matrix is uniquely expressible as the sum of a Symmetric matrix and a

 Skew-Symmetric matrix.

Let  A = P + Q , where  P  is Symmetric and  Q  is Skew-Symmetric.

∴  A' = ( P + Q )'= P ' + Q '  =  P + ( - Q ) = P - Q .

solving ,  we get     

∴ Representation  of  A as the sum