Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Explanation

Symmetric matrix is a matrix whose transpose is that matrix itself and skewsymmetric matrix is a matrix whose transpose is negative of that matrix.

Let `A` be any square matrix. Then,


`A=frac{1}{2}(A+A^{T})+frac{1}{2}(A-A^{T})=P+Q `


where, `P=frac{1}{2}(A+A^{T}), Q=frac{1}{2}(A-A^{T})`.

Now, `P^{T}=(frac{1}{2}(~A+A^{T}))^{T}=frac{1}{2}(~A+A^{T})^{T} [because(KT)^{T}=K cdot A^{T}]`


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