Question #12467

Prove that any matrix ring as vector space is direct sum of sets of symmetric and antisymmetric matrices.

Expert's answer

It is evident, that it is sufficiently to prove that any matrix A can be written as A1 + A2, where A1 is symmetric and A2 is antisymmetric matrices.

One can easily see that A1 = 1/2(A + A^{T} ) is symmetric and A2 = 1/2(A - A^{T} ). Moreover, A = A1 + A2. And that's it.

One can easily see that A1 = 1/2(A + A

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