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"("x"), or "f"("x"), is a compact subset ["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.
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 "λ"n is given by: 
Similarly, the minimal eigenvalue "λ"1 is given by: 
Counterexample in the non-Hermitian case.
Let "N" be the nilpotent matrix
Define the Rayleigh quotient formula_9 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 {"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,
Here formula_12 denotes the "k"th entry in the increasing sequence of σ's, so that formula_13.
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:
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 (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 as
where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write formula_18.) 
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.
 \max_{x \in S_{k-1}^{\perp}, \|x\|=1} (Ax, x) \ge \lambda_k.</math>
Pick "S""k"−1 = span{"u"1, ..., "u""k"−1} and we deduce
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.
formula_22.
If we only have "N" eigenvalues and hence run out of eigenvalues, then we let formula_23 (the bottom of the essential spectrum) for "n>N", and the above statement holds after replacing min-max with inf-sup.
formula_25.
If we only have "N" eigenvalues and hence run out of eigenvalues, then we let formula_23 (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:
formula_30
and
formula_31.