Change of coordinates and a matrix representation of an inner product

Question

I'm working in $\mathbb{C}^n$. Any inner product can be represented with a positive definite matrix $A$ such that $\left<v, w\right> = v^* A w$. Here I want to introduce a new basis. Let $Q$ be the matrix whose columns are the new basis vectors. I'd like to find a matrix $B$ that represents the original inner product in the new basis. How can I express $B$ in terms of $Q$?

Answer 1

You want a matrix $B$ such that $v^* A w = v_0^* B w_0$ where $v_0$ and $w_0$ are representations of $v$ and $w$, respectively, in the new basis. Using $v_0 = Q^{-1} v$ and $w_0 = Q^{-1} w$, we obtain $v^* A w = v^* (Q^{-1})^* B Q^{-1} w$. This should be satisfied for any $v$ and $w$. Thus $A = (Q^{-1})^* B Q^{-1}$ and this implies $$B = Q^* A Q.$$