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 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 A, the range of the continuous function R_A(x), or 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. The min-max theorem is a refinement of this fact.

Min-max theorem

Let ${\textstyle A}$ be Hermitian on an inner product space ${\textstyle V}$ with dimension ${\textstyle n}$, with spectrum ordered in descending order ${\textstyle \lambda _{1}\geq ...\geq \lambda _{n}}$.

Let ${\textstyle v_{1},...,v_{n}}$ be the corresponding unit-length orthogonal eigenvectors.

Reverse the spectrum ordering, so that ${\textstyle \xi _{1}=\lambda _{n},...,\xi _{n}=\lambda _{1}}$.

(Poincare’s inequality) — Let ${\textstyle M}$ be a subspace of ${\textstyle V}$ with dimension ${\textstyle k}$, then there exists unit vectors ${\textstyle x,y\in M}$, such that

${\textstyle \langle x,Ax\rangle \leq \lambda _{k}}$, and ${\textstyle \langle y,Ay\rangle \geq \xi _{k}}$.

Proof

Part 2 is a corollary, using ${\textstyle -A}$.

${\textstyle M}$ is a ${\textstyle k}$ dimensional subspace, so if we pick any list of ${\textstyle n-k+1}$ vectors, their span ${\textstyle N:=span(v_{k},...v_{n})}$ must intersect ${\textstyle M}$ on at least a single line.

Take unit ${\textstyle x\in M\cap N}$. That’s what we need.

${\textstyle x=\sum _{i=k}^{n}a_{i}v_{i}}$, since ${\textstyle x\in N}$.

Since ${\textstyle \sum _{i=k}^{n}|a_{i}|^{2}=1}$, we find ${\textstyle \langle x,Ax\rangle =\sum _{i=k}^{n}|a_{i}|^{2}\lambda _{i}\leq \lambda _{k}}$.

min-max theorem — 

$${\displaystyle {\begin{aligned}\lambda _{k}&=\max _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=k\end{array}}\min _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle \\&=\min _{\begin{array}{c}{\mathcal {M}}\subset V\\\operatorname {dim} ({\mathcal {M}})=n-k+1\end{array}}\max _{\begin{array}{c}x\in {\mathcal {M}}\\\|x\|=1\end{array}}\langle x,Ax\rangle {\text{. }}\end{aligned}}}$$

Proof

Part 2 is a corollary of part 1, by using ${\textstyle -A}$.

By Poincare’s inequality, ${\textstyle \lambda _{k}}$ is an upper bound to the right side.

By setting ${\textstyle {\mathcal {M}}=span(v_{1},...v_{k})}$, the upper bound is achieved.

Applications

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.

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

External links and citations to related work

• Fisk, Steve (2005). "A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices". arXiv:math/0502408. {{cite journal}}: Cite journal requires |journal= (help)
• Hwang, Suk-Geun (2004). "Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices". The American Mathematical Monthly. 111 (2): 157–159. doi:10.2307/4145217. JSTOR 4145217.
• Kline, Jeffery (2020). "Bordered Hermitian matrices and sums of the Möbius function". Linear Algebra and Its Applications. 588: 224–237. doi:10.1016/j.laa.2019.12.004.
• Reed, Michael; Simon, Barry (1978). Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press. ISBN 978-0-08-057045-7.