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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 $E$?

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  • $\begingroup$ Yes, for example when all eigenvalues are equal, that is when M is proportional to I. $\endgroup$
    – Alexandre Eremenko
    Commented Jun 17, 2020 at 19:29
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    $\begingroup$ As the Rayleigh quotient is scale invariant, you can restrict to unit vectors in $E$. This subspace of $E$ is compact and the Rayleigh quotient is continuous, so the Rayleigh quotient attains its max and min. $\endgroup$
    – Neal
    Commented Jun 17, 2020 at 19:32

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You can restrict the inf/sup to $\{x\in E : \|x\|=1\}$, which is compact, so this follows from the extreme value theorem.

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