Given a complex inner product on a finite-dimensional vector space, is there always a matrix $M$ such that $\langle x,y \rangle=y^*Mx$. What are the properties of such a matrix?
I saw on the wiki page that if the vector space is $\mathbb{C}^n$ then there is always such a matrix, which is Hermitian positive-definite. I was just wondering if the same could be said for other finite-dimensional spaces.
Given an inner product space $V$ with basis $B = \{v_1,\dots,v_n\}$, we can write $$ \langle x,y \rangle = [y]_B^* [M]_B[x]_B $$ Here, $[x]_B$ is the coordinate vector of $x$ with respect to the basis $B$. The matrix $[M]_B$ can be defined by $$ [M]_B = \pmatrix{ \langle v_1,v_1 \rangle & \cdots & \langle v_1,v_n \rangle\\ \vdots & \ddots & \vdots\\ \langle v_n,v_1 \rangle & \cdots & \langle v_n,v_n \rangle } $$ The matrix $[M]_B$ must be positive definite, and every positive definite matrix defines an inner product in this fashion.