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In "Numerical Linear Algebra" by Trefethan and Bau, there is the following question:

Let A $\in \mathbb{C}^{m\times m}$, not necessarily Hermitian, and $z$ is a number. Show that $z$ is a Rayleigh quotient of $A$ if and only if it is a diagonal entry of $Q^*AQ$ for some unitary matrix $Q$.

I don't really know where to start, but I'm thinking it has something to do with eigenvalues/eigenvectors of $A$. I know that $R[A,x] \in [\lambda_{\min},\lambda_{\max}]$ as a Rayleigh quotient is a weighted average of the eigenvalues of $A$. I think its the unitary matrices that are throwing me off here. I know that the Rayleigh quotient for a matrix $A$ and vector $x$ is defined as:

$$R(x) = \cfrac{x^*Ax}{x^*x}$$

Thanks for any help/suggestions.

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    – A.P.
    Commented Nov 5, 2015 at 12:04

1 Answer 1

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Hint: Consider a matrix $Q$ whose first row is $z^* = v_1^*$, so that $Q^*$ can be written as the columns $$ Q^* = [v_1\quad v_2\quad\cdots\quad v_n] $$ Consider the block-matrix product $QAQ^*$, and show that that the $i$th diagonal entry is $q_i^*Aq_i$.

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