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. 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 A be a n × n Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient RA : Cn \ {0} → R defined by
R A ( x ) = ( A x , x ) ( x , x ) {\displaystyle R_{A}(x)={\frac {(Ax,x)}{(x,x)}}}
where (⋅, ⋅) denotes the Euclidean inner product on Cn. The Rayleigh quotient of an eigenvector v {\displaystyle v} is its associated eigenvalue λ {\displaystyle \lambda } because R A ( v ) = ( λ x , x ) / ( x , x ) = λ {\displaystyle R_{A}(v)=(\lambda x,x)/(x,x)=\lambda } . 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 Let A {\textstyle A} be Hermitian on an inner product space V {\textstyle V} with dimension n {\textstyle n} , with spectrum ordered in descending order λ 1 ≥ . . . ≥ λ n {\textstyle \lambda _{1}\geq ...\geq \lambda _{n}} . Let v 1 , . . . , v n {\textstyle v_{1},...,v_{n}} be the corresponding unit-length orthogonal eigenvectors. Reverse the spectrum ordering, so that ξ 1 = λ n , . . . , ξ n = λ 1 {\textstyle \xi _{1}=\lambda _{n},...,\xi _{n}=\lambda _{1}} .
Define the partial trace t r V ( A ) {\textstyle tr_{V}(A)} to be the trace of projection of A {\textstyle A} to V {\textstyle V} . It is equal to ∑ i v i ∗ A v i {\textstyle \sum _{i}v_{i}^{*}Av_{i}} given an orthonormal basis of V {\textstyle V} .
This has some corollaries:
Counterexample in the non-Hermitian case Let N be the nilpotent matrix
[ 0 1 0 0 ] . {\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}
Define the Rayleigh quotient R N ( x ) {\displaystyle R_{N}(x)} 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 1/2. 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:
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