Originally Posted by

**redsoxfan325** The question reads:

*Let $\displaystyle x,y$ be vectors in $\displaystyle \mathbb{C}^n$, and assume that $\displaystyle x\neq0$. Prove that there is a symmetric matrix $\displaystyle B$ such that $\displaystyle Bx=y$.*

I don't really know where to begin on this. It's in the chapter on bilinear forms, but I don't see what this has to do with that. Maybe find some vector $\displaystyle z$ such that $\displaystyle z^{\top}y=0$, so that we have to solve $\displaystyle z^{\top}Bx=0$, in other words, show that there exists a vector $\displaystyle z$ that is orthogonal to $\displaystyle x$ with respect to the bilinear form $\displaystyle B$ (and then maybe use the fact that the eigenvectors of $\displaystyle B$ form an orthogonal basis).

But other than that speculation, I got nothing.

Help is appreciated!