non-singular transformation of a square matrix in an inner product

Question

Let $x$ be some vector. When $A$ is a square matrix, $A^*$ is the adjoint matrix of $A$, and $P$ is a non-singular matrix, is the inner product of $(A^{*}Ax,x)$ always identical to $(Ax,P^{-1}APx)$? If so, how can it be proved?

Answer 1

\begin{aligned} 1&=\left\langle\pmatrix{1&0\\ 0&0}^\ast\pmatrix{1&0\\ 0&0}\pmatrix{1\\ 0},\ \pmatrix{1\\ 0}\right\rangle\\ \ne0&=\left\langle\pmatrix{1&0\\ 0&0}\pmatrix{1\\ 0}, \ \pmatrix{0&1\\ 1&0}\pmatrix{1&0\\ 0&0}\pmatrix{0&1\\ 1&0}\pmatrix{1\\ 0}\right\rangle. \end{aligned}