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 Template:Mvar 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

$R_A(x)=\frac{(Ax,x)}{(x,x)}$

where (⋅, ⋅) denotes the Euclidean inner product on Cⁿ. Equivalently, the Rayleigh–Ritz quotient can be replaced by

$f(x)=(Ax,x),\Vert x\Vert =1.$

The Rayleigh quotient of an eigenvector $v$ is its associated eigenvalue $\lambda$ because $R_A(v)=(\lambda x,x)/(x,x)=\lambda$. For a Hermitian matrix A, the range of the continuous functions R_A(x) and 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 $A$ be Hermitian on an inner product space $V$ with dimension $n$, with spectrum ordered in descending order ${\lambda }_1\ge ...\ge {\lambda }_n$.

Let $v_1,...,v_n$ be the corresponding unit-length orthogonal eigenvectors.

Reverse the spectrum ordering, so that ${\xi }_1={\lambda }_n,...,{\xi }_n={\lambda }_1$.

Define the partial trace $t{r}_V(A)$ to be the trace of projection of $A$ to $V$. It is equal to $\sum _i{v}_i^{*}A{v}_i$ given an orthonormal basis of $V$.

Proof

Template:Math proof

This has some corollaries:Template:Pg Template:Math theorem

Proof

Template:Math proof

Counterexample in the non-Hermitian case

Let N be the nilpotent matrix

$\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right].$

Define the Rayleigh quotient $R_N(x)$ 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 Template:Sfrac. 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:

${\sigma }_k^{\downarrow }=\underset{S:\dim (S)=k}{\max }\underset{x\in S,\Vert x\Vert =1}{\min }(M^{*}Mx,x{)}^{\frac{1}{2}}=\underset{S:\dim (S)=k}{\max }\underset{x\in S,\Vert x\Vert =1}{\min }\Vert Mx\Vert .$

Similarly,

${\sigma }_k^{\downarrow }=\underset{S:\dim (S)=n-k+1}{\min }\underset{x\in S,\Vert x\Vert =1}{\max }\Vert Mx\Vert .$

Here ${\sigma }_k^{\downarrow }$ denotes the k^th entry in the decreasing sequence of the singular values, so that ${\sigma }_1^{\downarrow }\ge {\sigma }_2^{\downarrow }\ge \cdots$.

Cauchy interlacing theorem

Let Template:Mvar be a symmetric n × n matrix. The m × m matrix B, where m ≤ n, is called a compression of Template:Mvar if there exists an orthogonal projection P onto a subspace of dimension m such that PAP* = B. The Cauchy interlacing theorem states:

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

${\alpha }_j\le {\beta }_j\le {\alpha }_{n-m+j}.$

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

${\beta }_j=\underset{x\in S_j,\Vert x\Vert =1}{\max }(Bx,x)=\underset{x\in S_j,\Vert x\Vert =1}{\max }(PA{P}^{*}x,x)\ge \underset{S_j}{\min }\underset{x\in S_j,\Vert x\Vert =1}{\max }(A(P^{*}x),P^{*}x)={\alpha }_j.$

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

${\beta }_j=\underset{x\in S_{m-j+1},\Vert x\Vert =1}{\min }(Bx,x)=\underset{x\in S_{m-j+1},\Vert x\Vert =1}{\min }(PA{P}^{*}x,x)=\underset{x\in S_{m-j+1},\Vert x\Vert =1}{\min }(A(P^{*}x),P^{*}x)\le {\alpha }_{n-m+j},$

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.

Lidskii's inequality

Proof

Template:Math proof

Note that $\sum _i{\lambda }_i(A+B)=tr(A+B)=\sum _i{\lambda }_i(A)+{\lambda }_i(B)$. In other words, $\lambda (A+B)-\lambda (A)⪯\lambda (B)$ where $⪯$ means majorization. By the Schur convexity theorem, we then have

Compact operators

Let Template:Mvar be a compact, Hermitian operator on a Hilbert space H. Recall that the non-zero spectrum of such an operator consists of real eigenvalues with finite multiplicities whose only possible cluster point is zero. If Template:Mvar has infinitely many positive eigenvalues, they accumulate at zero. In this case, we list the positive eigenvalues of Template:Mvar as

$\cdots \le {\lambda }_k\le \cdots \le {\lambda }_1,$

where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write ${\lambda }_k={\lambda }_k^{\downarrow }$.) 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 Template:Mvar be a compact, self-adjoint operator on a Hilbert space Template:Mvar, whose positive eigenvalues are listed in decreasing order ... ≤ λ_k ≤ ... ≤ λ₁. Then:

$\begin{array}{rl}\underset{S_k}{\max }\underset{x\in S_k,\Vert x\Vert =1}{\min }(Ax,x)& ={\lambda }_k^{\downarrow },\\ \underset{S_{k-1}}{\min }\underset{x\in S_{k-1}^{\perp },\Vert x\Vert =1}{\max }(Ax,x)& ={\lambda }_k^{\downarrow }.\end{array}$

A similar pair of equalities hold for negative eigenvalues.

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.

Theorem (Min-Max). Let A be self-adjoint, and let $E_1\le E_2\le E_3\le \cdots$ be the eigenvalues of A below the essential spectrum. Then

$E_n=\underset{{\psi }_1,\dots ,{\psi }_n}{\min }\max \{⟨\psi ,A\psi ⟩:\psi \in \mathrm{span}({\psi }_1,\dots ,{\psi }_n),\Vert \psi \Vert =1\}$.

If we only have N eigenvalues and hence run out of eigenvalues, then we let $E_n:=\inf {\sigma }_{ess}(A)$ (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 $E_1\le E_2\le E_3\le \cdots$ be the eigenvalues of A below the essential spectrum. Then

$E_n=\underset{{\psi }_1,\dots ,{\psi }_{n-1}}{\max }\min \{⟨\psi ,A\psi ⟩:\psi \perp {\psi }_1,\dots ,{\psi }_{n-1},\Vert \psi \Vert =1\}$.

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

The proofs use the following results about self-adjoint operators:

Theorem. Let A be self-adjoint. Then $(A-E)\ge 0$ for $E\in \mathbb{ℝ}$ if and only if $\sigma (A)\subseteq [E,\mathrm{\infty })$.Template:Rp

Theorem. If A is self-adjoint, then

$\inf \sigma (A)=\underset{\psi \in \mathfrak{𝔇}(A),\Vert \psi \Vert =1}{\inf }⟨\psi ,A\psi ⟩$

and

$\sup \sigma (A)=\underset{\psi \in \mathfrak{𝔇}(A),\Vert \psi \Vert =1}{\sup }⟨\psi ,A\psi ⟩$.Template:Rp

See also

• Courant minimax principle
• Max–min inequality