Linear and multilinear algebra; matrix theory Eigenvalues and eigenvectors Theorem

Cauchy interlacing theorem

Builds on Hermitian matrix Eigenvalue Courant–Fischer min-max theorem

Statement

Let $1 \le m \le n$ , and let $A$ be an $n \times n$ Hermitian matrix with eigenvalues $\alpha_1 \ge \dots \ge \alpha_n$ . Let $P$ be an $n \times m$ complex matrix with $P^* P = I_m$ , and let $B = P^* A P$ , with eigenvalues $\beta_1 \ge \dots \ge \beta_m$ . Then, for $j = 1, \dots, m$ , $\alpha_j \ge \beta_j \ge \alpha_{j + n - m}.$ In particular, if $B$ is obtained from $A$ by deleting row $i$ and column $i$ for some $i$ , then $\alpha_1 \ge \beta_1 \ge \alpha_2 \ge \beta_2 \ge \dots \ge \alpha_{n-1} \ge \beta_{n-1} \ge \alpha_n.$

Remarks

Both inequalities follow from the Courant–Fischer min-max theorem. A subspace $W$ of $\C^m$ of dimension $j$ maps to the subspace $PW$ of $\C^n$ of dimension $j$ , and $P$ preserves the Rayleigh quotient: $R_B(w) = R_A(Pw)$ . So the max-min for $\beta_j$ runs over fewer subspaces than the one for $\alpha_j$ , which gives $\beta_j \le \alpha_j$ . The min-max form gives the other inequality.

Example: $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $3$ and $1$ . Deleting the second row and column leaves $(2)$ , and indeed $3 \ge 2 \ge 1$ .

Going deeper

For a student

Interlacing is used to bound eigenvalues of graphs: the adjacency matrix of an induced subgraph is a principal submatrix. Applied repeatedly, it compares the eigenvalues of $A$ with those of each of its principal submatrices. The converse holds too: any two interlacing sequences arise from a Hermitian matrix and a principal submatrix of size one less.

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Dimension concept
Eigenvalue concept
Hermitian matrix concept
Linear subspace concept
Rayleigh quotient concept
Courant–Fischer min-max theorem statement
Cauchy interlacing theorem statement

Tag	`8QQPVNR80F`
URI	`m2://8QQPVNR80F`
Kind	statement, theorem
Writing	W0 stub
MSC	15A42, 15A18
Status	This copy of the site has no build data. Run `m2 status 8QQPVNR80F`.
About	Eigenvalue, Hermitian matrix
Uses	Hermitian matrix, Eigenvalue, Courant–Fischer min-max theorem
Changed	2 November 2026