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.

Not to be confused with Minimax theorem.

"Variational theorem" redirects here. It is not to be confused with variational principle.

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 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

${\displaystyle R_{A}(x)={\frac {(Ax,x)}{(x,x)}}}$

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

${\displaystyle f(x)=(Ax,x),\;\|x\|=1.}$

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.

Min-max theorem

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

${\displaystyle \lambda _{k}=\min _{U}\{\max _{x}\{R_{A}(x)\mid x\in U{\text{ and }}x\neq 0\}\mid \dim(U)=k\}}$

and

${\displaystyle \lambda _{k}=\max _{U}\{\min _{x}\{R_{A}(x)\mid x\in U{\text{ and }}x\neq 0\}\mid \dim(U)=n-k+1\}}$

in particular,

${\displaystyle \lambda _{1}\leq R_{A}(x)\leq \lambda _{n}\quad \forall x\in \mathbf {C} ^{n}\backslash \{0\}}$

and these bounds are attained when x is an eigenvector of the appropriate eigenvalues.

Also the simpler formulation for the maximal eigenvalue λ_n is given by:

${\displaystyle \lambda _{n}=\max\{R_{A}(x):x\neq 0\}.}$

Similarly, the minimal eigenvalue λ₁ is given by:

${\displaystyle \lambda _{1}=\min\{R_{A}(x):x\neq 0\}.}$

Proof —

Since the matrix A is Hermitian it is diagonalizable and we can choose an orthonormal basis of eigenvectors {u₁, ..., u_n} that is, u_i is an eigenvector for the eigenvalue λ_i and such that (u_i, u_i) = 1 and (u_i, u_j) = 0 for all i ≠ j.

If U is a subspace of dimension k then its intersection with the subspace span{u_k, ..., u_n} isn't zero (by simply checking dimensions) and hence there exists a vector v ≠ 0 in this intersection that we can write as

${\displaystyle v=\sum _{i=k}^{n}\alpha _{i}u_{i}}$

and whose Rayleigh quotient is

${\displaystyle R_{A}(v)={\frac {\sum _{i=k}^{n}\lambda _{i}\alpha _{i}^{2}}{\sum _{i=k}^{n}\alpha _{i}^{2}}}\geq \lambda _{k}}$

(as all ${\displaystyle \lambda _{i}\geq \lambda _{k}}$ for i=k,..,n) and hence

${\displaystyle \max\{R_{A}(x)\mid x\in U\}\geq \lambda _{k}}$

Since this is true for all U, we can conclude that

${\displaystyle \min\{\max\{R_{A}(x)\mid x\in U{\text{ and }}x\neq 0\}\mid \dim(U)=k\}\geq \lambda _{k}}$

This is one inequality. To establish the other inequality, chose the specific k-dimensional space V = span{u₁, ..., u_k} , for which

${\displaystyle \max\{R_{A}(x)\mid x\in V{\text{ and }}x\neq 0\}\leq \lambda _{k}}$

because ${\displaystyle \lambda _{k}}$ is the largest eigenvalue in V. Therefore, also

${\displaystyle \min\{\max\{R_{A}(x)\mid x\in U{\text{ and }}x\neq 0\}\mid \dim(U)=k\}\leq \lambda _{k}}$

In the case where U is a subspace of dimension n-k+1, we proceed in a similar fashion: Consider the subspace of dimension k, span{u₁, ..., u_k}. Its intersection with the subspace U isn't zero (by simply checking dimensions) and hence there exists a vector v in this intersection that we can write as

${\displaystyle v=\sum _{i=1}^{k}\alpha _{i}u_{i}}$

and whose Rayleigh quotient is

${\displaystyle R_{A}(v)={\frac {\sum _{i=1}^{k}\lambda _{i}\alpha _{i}^{2}}{\sum _{i=1}^{k}\alpha _{i}^{2}}}\leq \lambda _{k}}$

and hence

${\displaystyle \min\{R_{A}(x)\mid x\in U\}\leq \lambda _{k}}$

Since this is true for all U, we can conclude that

${\displaystyle \max\{\min\{R_{A}(x)\mid x\in U{\text{ and }}x\neq 0\}\mid \dim(U)=n-k+1\}\leq \lambda _{k}}$

Again, this is one part of the equation. To get the other inequality, note again that the eigenvector u of ${\displaystyle \lambda _{k}}$ is contained in U = span{u_k, ..., u_n} so that we can conclude the equality.

Counterexample in the non-Hermitian case

Let N be the nilpotent matrix

${\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}$

Define the Rayleigh quotient ${\displaystyle R_{N}(x)}$ 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 ratio 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 {σ_k} 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:

${\displaystyle \sigma _{k}^{\uparrow }=\min _{S:\dim(S)=k}\max _{x\in S,\|x\|=1}(M^{*}Mx,x)^{\frac {1}{2}}=\min _{S:\dim(S)=k}\max _{x\in S,\|x\|=1}\|Mx\|.}$

Similarly,

${\displaystyle \sigma _{k}^{\uparrow }=\max _{S:\dim(S)=n-k+1}\min _{x\in S,\|x\|=1}\|Mx\|.}$

Here ${\displaystyle \sigma _{k}=\sigma _{k}^{\uparrow }}$ denotes the k^th entry in the increasing sequence of σ's, so that ${\displaystyle \sigma _{1}\leq \sigma _{2}\leq \cdots }$.

Cauchy interlacing theorem

Main article: Poincaré separation theorem

Let A be a symmetric n × n matrix. 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 P*AP = B. The Cauchy interlacing theorem states:

Theorem. If the eigenvalues of A are α₁ ≤ ... ≤ α_n, and those of B are β₁ ≤ ... ≤ β_j ≤ ... ≤ β_m, then for all j ≤ m,

${\displaystyle \alpha _{j}\leq \beta _{j}\leq \alpha _{n-m+j}.}$

This can be proven using the min-max principle. Let β_i have corresponding eigenvector b_i and S_j be the j dimensional subspace S_j = span{b₁, ..., b_j}, then

${\displaystyle \beta _{j}=\max _{x\in S_{j},\|x\|=1}(Bx,x)=\max _{x\in S_{j},\|x\|=1}(P^{*}APx,x)\geq \min _{S_{j}}\max _{x\in S_{j},\|x\|=1}(A(Px),Px)=\alpha _{j}.}$

According to first part of min-max, α_j ≤ β_j. On the other hand, if we define S_m−j+1 = span{b_j, ..., b_m}, then

${\displaystyle \beta _{j}=\min _{x\in S_{m-j+1},\|x\|=1}(Bx,x)=\min _{x\in S_{m-j+1},\|x\|=1}(P^{*}APx,x)=\min _{x\in S_{m-j+1},\|x\|=1}(A(Px),Px)\leq \alpha _{n-m+j},}$

where the last inequality is given by the second part of min-max.

When n − m = 1, we have α_j ≤ β_j ≤ α_j+1, hence the name interlacing theorem.

Compact operators

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

${\displaystyle \cdots \leq \lambda _{k}\leq \cdots \leq \lambda _{1},}$

where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write ${\displaystyle \lambda _{k}=\lambda _{k}^{\downarrow }}$.) 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 S_k ⊂ H be a k dimensional subspace, we can obtain the following theorem.

Theorem (Min-Max). Let A be a compact, self-adjoint operator on a Hilbert space H, whose positive eigenvalues are listed in decreasing order ... ≤ λ_k ≤ ... ≤ λ₁. Then:

${\displaystyle {\begin{aligned}\max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow },\\\min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)&=\lambda _{k}^{\downarrow }.\end{aligned}}}$

A similar pair of equalities hold for negative eigenvalues.

Proof —

Let S' be the closure of the linear span ${\displaystyle S'=\operatorname {span} \{u_{k},u_{k+1},\ldots \}}$. The subspace S' has codimension k − 1. By the same dimension count argument as in the matrix case, S' ∩ S_k is non empty. So there exists x ∈ S'  ∩ S_k with ${\displaystyle \|x\|=1}$. Since it is an element of S' , such an x necessarily satisfy

${\displaystyle (Ax,x)\leq \lambda _{k}.}$

Therefore, for all S_k

${\displaystyle \inf _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}}$

But A is compact, therefore the function f(x) = (Ax, x) is weakly continuous. Furthermore, any bounded set in H is weakly compact. This lets us replace the infimum by minimum:

${\displaystyle \min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}$

So

${\displaystyle \sup _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)\leq \lambda _{k}.}$

Because equality is achieved when ${\displaystyle S_{k}=\operatorname {span} \{u_{1},\ldots ,u_{k}\}}$,

${\displaystyle \max _{S_{k}}\min _{x\in S_{k},\|x\|=1}(Ax,x)=\lambda _{k}.}$

This is the first part of min-max theorem for compact self-adjoint operators.

Analogously, consider now a (k − 1)-dimensional subspace S_k−1, whose the orthogonal complement is denoted by S_k−1^⊥. If S' = span{u₁...u_k},

${\displaystyle S'\cap S_{k-1}^{\perp }\neq {0}.}$

So

${\displaystyle \exists x\in S_{k-1}^{\perp }\,\|x\|=1,(Ax,x)\geq \lambda _{k}.}$

This implies

${\displaystyle \max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}}$

where the compactness of A was applied. Index the above by the collection of k-1-dimensional subspaces gives

${\displaystyle \inf _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)\geq \lambda _{k}.}$

Pick S_k−1 = span{u₁, ..., u_k−1} and we deduce

${\displaystyle \min _{S_{k-1}}\max _{x\in S_{k-1}^{\perp },\|x\|=1}(Ax,x)=\lambda _{k}.}$

Self-adjoint operators

The min-max theorem also applies to (possibly unbounded) self-adjoint operators.[1][2] 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 self-adjoint, and let ${\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }$ be the eigenvalues of A below the essential spectrum. Then

${\displaystyle E_{n}=\min _{\psi _{1},\ldots ,\psi _{n}}\max\{\langle \psi ,A\psi \rangle$ :\psi \in \operatorname {span} (\psi _{1},\ldots ,\psi _{n}),\,\|\psi \|=1\}} .

If we only have N eigenvalues and hence run out of eigenvalues, then we let ${\displaystyle E_{n}:=\inf \sigma _{ess}(A)}$ (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 self-adjoint, and let ${\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots }$ be the eigenvalues of A below the essential spectrum. Then

${\displaystyle E_{n}=\max _{\psi _{1},\ldots ,\psi _{n-1}}\min\{\langle \psi ,A\psi \rangle$ :\psi \perp \psi _{1},\ldots ,\psi _{n-1},\,\|\psi \|=1\}} .

If we only have N eigenvalues and hence run out of eigenvalues, then we let ${\displaystyle E_{n}:=\inf \sigma _{ess}(A)}$ (the bottom of the essential spectrum) for n > N, and the above statement holds after replacing max-min with sup-inf.

The proofs[1][2] use the following results about self-adjoint operators:

Theorem. Let A be self-adjoint. Then ${\displaystyle (A-E)\geq 0}$ for ${\displaystyle E\in \mathbb {R} }$ if and only if ${\displaystyle \sigma (A)\subseteq [E,\infty )}$.[1]^:77

Theorem. If A is self-adjoint, then

${\displaystyle \inf \sigma (A)=\inf _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }$

and

${\displaystyle \sup \sigma (A)=\sup _{\psi \in {\mathfrak {D}}(A),\|\psi \|=1}\langle \psi ,A\psi \rangle }$.[1]^:77

See also

• Courant minimax principle
• Max–min inequality

References

1. G. Teschl, Mathematical Methods in Quantum Mechanics (GSM 99) https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf
2. Lieb; Loss (2001). Analysis. GSM. 14 (2nd ed.). Providence: American Mathematical Society. ISBN 0-8218-2783-9.

• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.