Prove that those subspaces are $T_a-invariants$.

Question

Prove that, for $A ∈ O_n$, the subspaces of symmetric and anti-symmetric matrices ("$M$") of $n$ order are $T_a-invariants$.

where $T_A |= A^{-1}MA$ .

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So... ...the definition of $T_A$ implicates that every $M$ is diagonalizable for an opportune $A$?

Wikipedia says this is the spectral theorem :P...

for the definition of $O_n$ i can say $T_A= ^tAMA$ Nothing else i can say :/

Answer 1

Hint: If if $A\in O_n$ then $A^{-1}=A^T$. So if $M$ is symmetric or anti-symmetric, what does that say about the relationship between $A^{-1}MA$ and $(A^{-1}MA)^T$?