Answer to Question #12467 in Linear Algebra for john.george.milnor

Question #12467
Prove that any matrix ring as vector space is direct sum of sets of symmetric and antisymmetric matrices.
1
Expert's answer
2012-08-09T08:30:48-0400
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 + AT ) is symmetric and A2 = 1/2(A - AT ). Moreover, A = A1 + A2. And that's it.

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