(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that every square real matrix X can be written in a unique way as the sum of a symmetric matrix A and a skew-symmetric matrix B.

2. Relevant equations

X = A + B

A = [tex]\frac{X+X^{T}}{2}[/tex]

B = [tex]\frac{X-X^{T}}{2}[/tex]

X = [tex]\frac{X+X^{T}}{2}[/tex] + [tex]\frac{X-X^{T}}{2}[/tex]

3. The attempt at a solution

So I tried to solve [tex]\frac{X+X^{T}}{2}[/tex] + [tex]\frac{X-X^{T}}{2}[/tex] and it gives out X as a solution. However, how can I know that A is a symmetric and B is a skew-symmetric? Any idea?

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# Homework Help: Prove the theorem for the matrix

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