For a function $f(A)$ where $A$ is a matrix prove that the following.

Question

I have to prove that $f(P^{-1} AP) = P^{-1} f(A) P$, where $A$ is a symmetric matrix and $$ f(x)=3x^2+2 $$

So far I have done the following:

Please verify my approach. Also is there an alternative proof? Thanks

Answer 1

So it is not any function, but a specific function. And the only part missing is $$g(P^{-1}AP) = P^{-1}g(A)P$$ for $g(x)=x^2$. Is that correct?

In detail we have: \begin{align*} P^{-1}g(A)P &= P^{-1}A^2P \\ &= P^{-1}AAP\\ &= P^{-1}AIAP\\ &= P^{-1}APP^{-1}AP\\ &= (P^{-1}AP)^2\\ &=g(P^{-1}AP) \end{align*}