---
tag: CJK2HS2508
kind: statement
role: theorem
name: Courant–Fischer min-max theorem
msc: [15A42, 15A18]
uses: [0P17NGQQMQ, 0BQAA2BK3Q, 06WHP8SE98, 00YC1D652K, 03GSZ8RTZJ]
about: [0BQAA2BK3Q, 0P17NGQQMQ]
sources:
  - url: https://en.wikipedia.org/wiki/Min-max_theorem
---

## Statement

Let $A$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$, listed with multiplicity.
For $x \in \C^n$ with $x \ne 0$, let $R_A(x) = \dfrac{x^* A x}{x^* x}$ be the [[Rayleigh quotient|06WHP8SE98]].
Then, for each $k = 1, \dots, n$,
$$\lambda_k = \max_{\dim U = k} \; \min_{x \in U \setminus \set{0}} R_A(x) = \min_{\dim U = n-k+1} \; \max_{x \in U \setminus \set{0}} R_A(x),$$
where $U$ runs over the [[linear subspaces|00YC1D652K]] of $\C^n$ of the given [[dimension|03GSZ8RTZJ]].

## Remarks

Let $u_1, \dots, u_n$ be an orthonormal basis of eigenvectors with $A u_i = \lambda_i u_i$.
On the span of $u_1, \dots, u_k$, the Rayleigh quotient is at least $\lambda_k$, so the max-min is at least $\lambda_k$.
Every subspace $U$ of dimension $k$ meets the span of $u_k, \dots, u_n$ in a nonzero vector, by counting dimensions, and there $R_A \le \lambda_k$.
So the max-min is exactly $\lambda_k$; the min-max form is proved in the same way.

For $k = 1$ it gives $\lambda_1 = \max_{x \ne 0} R_A(x)$, and for $k = n$ it gives $\lambda_n = \min_{x \ne 0} R_A(x)$.
For $A = \operatorname{diag}(3, 1)$, $R_A(x) = (3x_1^2 + x_2^2)/(x_1^2 + x_2^2)$ for real $x$, which takes all values in $[1, 3]$.

The Hermitian hypothesis is needed: for a non-Hermitian matrix the eigenvalues need not be real, and $R_A$ need not be real either.

## Explanations

### L1 human

The eigenvalues of a symmetric matrix can be found by an optimisation, without solving any polynomial equation.
The largest eigenvalue is the largest value of the Rayleigh quotient.
The second one is the best value one can guarantee on every plane, and so on.

### L2 human

The theorem turns questions about eigenvalues into questions about subspaces.
It gives the [[Cauchy interlacing theorem|8QQPVNR80F]] and [[Weyl's inequality|RACSNQXDQV]] in a few lines.
The real symmetric case is the same, with $\R^n$ and $x^T A x$.
A version holds for compact self-adjoint operators on Hilbert spaces.