nLab Cauchy interlace theorem

Statement

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

Then for every principal submatrix $B \in Mat_{n \times n }(\mathbb{C})$, obtained by deleting the $i$th column and the $i$th row from $A$, for some $i \in \{1,\cdots, n+1\}$, the eigenvalues $(b_1 \leq b_2 \leq \cdots b_n)$ of $B$ interlace the eigenvalues $(a_1 \leq a_2 \leq \cdots a_{n+1})$ of $A$, in that

$$a_1 \;\leq\; b_1 \;\leq\; a_2 \;\leq\; b_2 \;\leq\; \cdots \;\leq\; b_n \;\leq\; a_{n+1} \,.$$

Related concepts

• principal submatrix

• Gershgorin circle theorem

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:

• Wikipedia, Cauchy interlacing theorem

• Wikipedia, Poincaré separation theorem