Exercise: Let $V = R^n$ with a non-standard inner product and let L be the multiplication by a matrix A. Find the matrix of $L^A$ (where $L^A$ is the adjoint operator) in terms of the matrix A and the metric matrix G.
My attempt: $<A|L(B)> = A^AGL(B) = (L(B)^AG^AA)^A$.
Now $A^AGL(B) = (L(B)^AG^AA)^A$ if and only if
$(A^AGL(B))^A = L(B)^AG^AA$ if and only if
$L(B)^A = (A^AGL(B))^A(G^AA)^-1$.
The problem is that on both sides of the equation L(B) is present, whereas I only want it on the left hand side. I actually tried to rewrite it such that $<A|L(B)>=<L(A)^A|B>$, but did not succeed. Any help, whether it is answers or hints, are appreciated.
Let $\cdot^G$ denote the adjoint w.r.t. your metric $G$ and let $\cdot^*$ denote the adjoint w.r.t. standard inner product. We have $$\langle x, A^*G^*y\rangle = \langle GAx,y\rangle = \langle Ax,y\rangle_G = \langle x,A^Gy\rangle_G = \langle Gx,A^Gy\rangle = \langle x, G^*A^Gy\rangle$$ for all $x,y \in \Bbb{R}^n$ so $$G^*A^G = A^*G^* \implies A^G = (G^*)^{-1}A^*G^* = (G^{-1})^*A^*G^*.$$ Note that $G$ is indeed invertible since $\langle \cdot, \cdot\rangle_G$ is an inner product.