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In linear algebra and functional analysis, the **min-max theorem**, or **variational theorem**, or **Courant–Fischer–Weyl min-max principle**, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.

This article first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument.

In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below.

Let A be a *n* × *n* Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient *R _{A}* :

where (⋅, ⋅) denotes the Euclidean inner product on **C**^{n}.
Clearly, the Rayleigh quotient of an eigenvector is its associated eigenvalue. Equivalently, the Rayleigh–Ritz quotient can be replaced by

For Hermitian matrices *A*, the range of the continuous function *R _{A}*(

Let be Hermitian on an inner product space with dimension , with spectrum ordered in descending order .

Let be the corresponding unit-length orthogonal eigenvectors.

Reverse the spectrum ordering, so that .

**(Poincaré’s inequality)** — Let be a subspace of with dimension , then there exists unit vectors , such that

, and .

Part 2 is a corollary, using .

is a dimensional subspace, so if we pick any list of vectors, their span must intersect on at least a single line.

Take unit . That’s what we need.

- , since .

- Since , we find .

**min-max theorem** —

Part 2 is a corollary of part 1, by using .

By Poincare’s inequality, is an upper bound to the right side.

By setting , the upper bound is achieved.

Let *N* be the nilpotent matrix

Define the Rayleigh quotient exactly as above in the Hermitian case. Then it is easy to see that the only eigenvalue of *N* is zero, while the maximum value of the Rayleigh quotient is 1/2. That is, the maximum value of the Rayleigh quotient is larger than the maximum eigenvalue.

The singular values {*σ _{k}*} of a square matrix

Similarly,

Here denotes the *k*^{th} entry in the increasing sequence of σ's, so that .

Main article: Poincaré separation theorem |

Let A be a symmetric *n* × *n* matrix. The *m* × *m* matrix *B*, where *m* ≤ *n*, is called a **compression** of A if there exists an orthogonal projection *P* onto a subspace of dimension *m* such that *PAP** = *B*. The Cauchy interlacing theorem states:

**Theorem.**If the eigenvalues of A are*α*_{1}≤ ... ≤*α*, and those of_{n}*B*are*β*_{1}≤ ... ≤*β*≤ ... ≤_{j}*β*, then for all_{m}*j*≤*m*,

This can be proven using the min-max principle. Let *β _{i}* have corresponding eigenvector

According to first part of min-max, *α _{j}* ≤

where the last inequality is given by the second part of min-max.

When *n* − *m* = 1, we have *α _{j}* ≤

Let A be a compact, Hermitian operator on a Hilbert space *H*. Recall that the spectrum of such an operator (the set of eigenvalues) is a set of real numbers whose only possible cluster point is zero.
It is thus convenient to list the positive eigenvalues of A as

where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write .)
When *H* is infinite-dimensional, the above sequence of eigenvalues is necessarily infinite.
We now apply the same reasoning as in the matrix case. Letting *S _{k}* ⊂

**Theorem (Min-Max).**Let A be a compact, self-adjoint operator on a Hilbert space H, whose positive eigenvalues are listed in decreasing order ... ≤*λ*≤ ... ≤_{k}*λ*_{1}. Then:

A similar pair of equalities hold for negative eigenvalues.

Let *S' * be the closure of the linear span .
The subspace *S' * has codimension *k* − 1. By the same dimension count argument as in the matrix case, *S' * ∩ *S _{k}* has positive dimension. So there exists

Therefore, for all *S _{k}*

But A is compact, therefore the function *f*(*x*) = (*Ax*, *x*) is weakly continuous. Furthermore, any bounded set in *H* is weakly compact. This lets us replace the infimum by minimum:

So

Because equality is achieved when ,

This is the first part of min-max theorem for compact self-adjoint operators.

Analogously, consider now a (*k* − 1)-dimensional subspace *S*_{k−1}, whose the orthogonal complement is denoted by *S*_{k−1}^{⊥}. If *S' * = span{*u*_{1}...*u _{k}*},

So

This implies

where the compactness of *A* was applied. Index the above by the collection of *k-1*-dimensional subspaces gives

Pick *S*_{k−1} = span{*u*_{1}, ..., *u*_{k−1}} and we deduce

The min-max theorem also applies to (possibly unbounded) self-adjoint operators.^{[1]}^{[2]} Recall the essential spectrum is the spectrum without isolated eigenvalues of finite multiplicity.
Sometimes we have some eigenvalues below the essential spectrum, and we would like to approximate the eigenvalues and eigenfunctions.

**Theorem (Min-Max).**Let*A*be self-adjoint, and let be the eigenvalues of*A*below the essential spectrum. Then

.

If we only have *N* eigenvalues and hence run out of eigenvalues, then we let (the bottom of the essential spectrum) for *n>N*, and the above statement holds after replacing min-max with inf-sup.

**Theorem (Max-Min).**Let*A*be self-adjoint, and let be the eigenvalues of*A*below the essential spectrum. Then

.

If we only have *N* eigenvalues and hence run out of eigenvalues, then we let (the bottom of the essential spectrum) for *n > N*, and the above statement holds after replacing max-min with sup-inf.

The proofs^{[1]}^{[2]} use the following results about self-adjoint operators:

**Theorem.**Let*A*be self-adjoint. Then for if and only if .^{[1]}^{: 77 }

**Theorem.**If*A*is self-adjoint, then

and

.^{[1]}^{: 77 }