- #1
frostshoxx
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Homework Statement
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.
Homework 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]
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?