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I am looking for a reference for the following result for symmetric matrices

Let $A\in\mathbb R^{n\times n}$ be symmetric with eigenvalues $\lambda_n \leq\ldots\leq\lambda_1,\, M\subset \lbrace 1 \ldots n\rbrace,\, |M|=n-1$ Now let $B\in\mathbb R^{n-1\times n-1},\, B=A(M)$ be the matrix $A$ without the $i$th row and column, where $i\notin M$

For the eigenvalues $\mu_{n-1} \leq \ldots \leq\mu_1$ of $B$ it holds

$$\lambda_n\leq \mu_{n-1}\leq \lambda_{n-1}\leq\ldots\leq\mu_1\leq\lambda_1 $$

Does anyone know who has proved this first? And maybe a link to the original paper?

user127.0.0.1
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2 Answers2

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i have heard the name "Cauchy interlacing theorem" (as well as "Cauchy's interlace theorem" or "Interlacing eigenvalues theorem for bordered matrices")Infor that theorem. that could be a hint that cauchy did it first or was among the first. sorry i have no idea where to find an original paper of that, but i hope my post helps anyways.

supinf
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It is an application of Courat-Fischer Theorem (or Min-Max Theorem), and here there is a problem on this subject with it's answer

Hamou
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