Let $M$ be an $n$ by $n$ real matrix (not necessarily symmetric), suppose $M$ is positive definite in that for any real vector $x \neq 0$, $x^\top M x > 0$. I'm wondering if $\min_{x\neq 0} \frac{x^\top M x}{x^\top x}$ has anything to do with the eigenvalues of $M$.
If $M$ is symmetric, I think it's the minimum eigenvalue. In general, I think it's about the relationship between the eigenvalues of $M$ and $M+M^\top$, this post https://mathoverflow.net/questions/52578/eigenvalues-of-sum-of-a-non-symmetric-matrix-and-its-transpose-aat seems relevant.