Given a positive definite matrix A, let $\langle x, y\rangle_A=x^\top Ay$. This inner product induces a norm $\|x\|_A^2=\langle x, x\rangle_A = x^\top A x$. My question is, what is the dual norm of $\|\cdot\|_A$?
The goal is to get something like a Holder's inequality:
$$
|x^\top y|\leq \|x\|_A \|y\|_{A, *}
$$
Note the LHS is the inner product in Euclidean space. Thanks a lot for any suggestions.
Best Answer
Let $A = L L^T$ be the Cholesky decomposition of $A$. Note that $$ x^T A x = 1 \iff \| y \|_2^2 = 1 $$ where $y = L^T x$.
The dual norm is \begin{align} \| z \|_* &= \sup_{x^T Ax = 1} \langle z, x \rangle \\ &= \sup_{\|y\|_2 = 1} \langle z, L^{-T}y \rangle \\ &= \sup_{\|y\|_2 = 1} \langle L^{-1} z, y \rangle \\ &= \| L^{-1} z \|_2 \\ &= \sqrt{z^T L^{-T} L^{-1} z } \\ &= \sqrt{z^T A^{-1} z}\\ &= \|z\|_{A^{-1}}\quad . \end{align}