Let $V=M_n\left(\mathbb{R}\right)$ and Let $P\in V$ be some invertible real matrix.
We define the operator $T:V\to V$ by: $\ \forall A\in V,\ T(A)=P^{-1}AP$.
The adjoint operator $T^*$ w.r.t the standard inner product $\left<A,B\right>=tr(B^tA)$ is given by $\forall A\in V, T^*(A)=\left(P^{-1}\right)^{t}AP^t$.
What is a necessary and sufficient condition for $T$ to be self-adjoint?
I can show that it is sufficient for $P$ to be self-adjoint, and that it is necessary for $P$ to be normal. But I am stuck in showing that one of these is both necessary and sufficient..
Hint: Note that if $P^{-1}AP = Q^{-1}AQ$ for all matrices $A$, then it follows that $$ A(PQ^{-1}) = (PQ^{-1})A $$ for all matrices $A$. It follows that $PQ^{-1}$ is a multiple of the identity matrix.