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.
For a Hermitian matrix A, the range of the continuous functions RA(x) and f(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.
Define a partial flag to be a nested collection
Let N be the nilpotent matrix Define the Rayleigh quotient
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.
An immediate consequence[citation needed] of the first equality in the min-max theorem is: Similarly, Here
denotes the kth entry in the decreasing sequence of the singular values, so that
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: This can be proven using the min-max principle.
Let βi have corresponding eigenvector bi and Sj be the j dimensional subspace Sj = span{b1, ..., bj}, then According to first part of min-max, αj ≤ βj.
On the other hand, if we define Sm−j+1 = span{bj, ..., bm}, then where the last inequality is given by the second part of min-max.
By the Schur convexity theorem, we then have p-Wielandt-Hoffman inequality —
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.
Letting Sk ⊂ H be a k dimensional subspace, we can obtain the following theorem.
A similar pair of equalities hold for negative eigenvalues.
Since it is an element of S' , such an x necessarily satisfy Therefore, for all Sk But A is compact, therefore the function f(x) = (Ax, x) is weakly continuous.
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 Sk−1, whose the orthogonal complement is denoted by Sk−1⊥.
If S' = span{u1...uk}, So This implies where the compactness of A was applied.
Index the above by the collection of k-1-dimensional subspaces gives Pick Sk−1 = span{u1, ..., uk−1} and we deduce The min-max theorem also applies to (possibly unbounded) self-adjoint operators.
(the bottom of the essential spectrum) for n>N, and the above statement holds after replacing min-max with inf-sup.
(the bottom of the essential spectrum) for n > N, and the above statement holds after replacing max-min with sup-inf.