Min-max theorem


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.

Matrices

Let be a Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh-Ritz quotient defined by
where denotes the Euclidean inner product on.
Clearly, the Rayleigh quotient of an eigenvector is its associated eigenvalue. Equivalently, the Rayleigh-Ritz quotient can be replaced by
For Hermitian matrices, the range of the continuous function RA, or f, is a compact subset of the real line. The maximum b and the minimum a are the largest and smallest eigenvalue of A, respectively. The min-max theorem is a refinement of this fact.

Min-max theorem

Let be an Hermitian matrix with eigenvalues then
and
in particular,
and these bounds are attained when is an eigenvector of the appropriate eigenvalues.
Also the simpler formulation for the maximal eigenvalue λn is given by:
Similarly, the minimal eigenvalue λ1 is given by:

Counterexample in the non-Hermitian case

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 ratio is. That is, the maximum value of the Rayleigh quotient is larger than the maximum eigenvalue.

Applications

Min-max principle for singular values

The singular values of a square matrix M are the square roots of the eigenvalues of M*M. An immediate consequence of the first equality in the min-max theorem is:
Similarly,
Here denotes the kth entry in the increasing sequence of σ's, so that.

Cauchy interlacing theorem

Let be a symmetric n × n matrix. The m × m matrix B, where m ≤ n, is called a compression of if there exists an orthogonal projection P onto a subspace of dimension m such that P*AP = B. The Cauchy interlacing theorem states:
This can be proven using the min-max principle. Let βi have corresponding eigenvector bi and Sj be the j dimensional subspace then
According to first part of min-max, On the other hand, if we define then
where the last inequality is given by the second part of min-max.
When, we have, hence the name interlacing theorem.

Compact operators

Let be a compact, Hermitian operator on a Hilbert space H. Recall that the spectrum of such an operator is a set of real numbers whose only possible cluster point is zero.
It is thus convenient to list the positive eigenvalues of as
where entries are repeated with multiplicity, as in the matrix case.
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 Sk ⊂ H be a k dimensional subspace, we can obtain the following theorem.
A similar pair of equalities hold for negative eigenvalues.

Self-adjoint operators

The min-max theorem also applies to self-adjoint operators. 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.
If we only have N eigenvalues and hence run out of eigenvalues, then we let for n>N, and the above statement holds after replacing min-max with inf-sup.
If we only have N eigenvalues and hence run out of eigenvalues, then we let for n > N, and the above statement holds after replacing max-min with sup-inf.
The proofs use the following results about self-adjoint operators:
and