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.

Let A be a n × n Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient R_A : Cⁿ \ {0} → R defined by

where (⋅, ⋅) denotes the Euclidean inner product on Cⁿ. 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 R_A(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.

Let A be a n × n Hermitian matrix with eigenvalues λ₁ ≤ ... ≤ λ_k ≤ ... ≤ λ_n then

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