Let $M\in\mathbb{C}^{n\times n}$ be a Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
$$\mbox{Are } \sup\_{x\in E\\
x\neq0}\dfrac{x^\*Mx}{x^\*x}\mbox{ and }\inf\_{x\in E\\
x\neq0}\dfrac{x^\*Mx}{x^\*x} \mbox{ always attained by some vector }x\in E, x\neq0 ?$$
If $\dim(E)=1$, then the answer would be yes, as the quotient will be constant for any $x\in E, x\neq0$.


If $\dim(E)=n$, then $E=\mathbb{C}^n$ and the max and min values of the Rayleigh quotient correspond to the largest and smallest eigenvalues of $M$.


Is the answer positive for any subspace $E$?



Can we attain the maximum and minimum of a Rayleigh quotient over any subspace?
You can restrict the inf/sup to $\{x\in E : \|x\|=1\}$, which is compact, so this follows from the extreme value theorem.


