Min-max theorem on arvau

Wikipedia / Min-max theorem

In linear algebra and functional analysis, the min-max theorem, or variational theorem, or CourantFischerWeyl 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. 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 .

. For a Hermitian matrix A, the range of the continuous functions RA(x) is a compact interval [a, b] 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

be the corresponding unit-length orthogonal eigenvectors.

Define the partial trace.

This has some corollaries:

Counterexample in the non-Hermitian case

Let N be the nilpotent matrix

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 . 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 {σk} of a square matrix M are the square roots of the eigenvalues of M*M (equivalently MM*). An immediate consequence of the first equality in the min-max theorem is:

Similarly,

denotes the kth.

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 PAP* = B. The Cauchy interlacing theorem states:

Theorem. If the eigenvalues of are , and those of B are , then for all ,

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.

Lidskii's inequality

means majorization. By the Schur convexity theorem, we then have

Compact operators

Let be a compact, Hermitian operator on a Hilbert space H. Recall that the non-zero spectrum of such an operator consists of real eigenvalues with finite multiplicities whose only possible cluster point is zero. If has infinitely many positive eigenvalues, they accumulate at zero. In this case, we list the positive eigenvalues of as

where entries are repeated with multiplicity.) 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.

Theorem (Min-Max). Let be a compact, self-adjoint operator on a Hilbert space , whose positive eigenvalues are listed in decreasing order . Then:

A similar pair of equalities hold for negative eigenvalues.

Self-adjoint operators

The min-max theorem also applies to (possibly unbounded) 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.

Theorem (Min-Max). Let A be the eigenvalues of A below the essential spectrum. Then

If we only have N (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 the eigenvalues of A below the essential spectrum. Then

If we only have N (the bottom of the essential spectrum) 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:

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

and