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Cauchy interlace theorem

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Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

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Homotopy groups

Basic facts

Theorems

Contents

Statement

For nn \in \mathbb{N}, let AMat n+1×n+1()A \in Mat_{n+1 \times n +1}(\mathbb{C}) be a hermitian matrix (for instance a symmetric matrix of all entries are real numbers).

Then for every principal submatrix BMat n×n()B \in Mat_{n \times n }(\mathbb{C}), obtained by deleting the iith column and the iith row from AA, for some i{1,,n+1}i \in \{1,\cdots, n+1\}, the eigenvalues (b 1b 2b n)(b_1 \leq b_2 \leq \cdots b_n) of BB interlace the eigenvalues (a 1a 2a n+1)(a_1 \leq a_2 \leq \cdots a_{n+1}) of AA, in that

a 1b 1a 2b 2b na n+1. a_1 \;\leq\; b_1 \;\leq\; a_2 \;\leq\; b_2 \;\leq\; \cdots \;\leq\; b_n \;\leq\; a_{n+1} \,.

References

  • Suk-Geun Hwang, Cauchy’s Interlace Theorem for Eigenvalues of Hermitian Matrices, The American Mathematical Monthly Vol. 111, No. 2 (Feb., 2004), pp. 157-159 (jstor:4145217)

  • Steve Fisk, A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian matrices (arXiv:math/0502408)

See also:

Last revised on April 29, 2021 at 05:48:30. See the history of this page for a list of all contributions to it.