Min-max theorem

Not to be confused with Minimax theorem.

"Variational theorem" redirects here. Not to be confused with variational principle.

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

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

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

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

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 ${\textstyle A}$ 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 ${\textstyle \xi _{1}=\lambda _{n},...,\xi _{n}=\lambda _{1}}$.

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

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.

Define the partial trace ${\textstyle tr_{V}(A)}$ to be the trace of projection of ${\textstyle A}$ to ${\textstyle V}$. It is equal to ${\textstyle \sum _{i}v_{i}^{*}Av_{i}}$ given an orthonormal basis of ${\textstyle V}$.

Wielandt minimax formula () — Let ${\textstyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ be integers. Define a partial flag to be a nested collection ${\textstyle V_{1}\subset \cdots \subset V_{k}}$ of subspaces of ${\textstyle \mathbb {C} ^{n}}$ such that ${\textstyle \operatorname {dim} \left(V_{j}\right)=i_{j}}$ for all ${\textstyle 1\leq j\leq k}$.

Define the associated Schubert variety ${\textstyle X\left(V_{1},\ldots ,V_{k}\right)}$ to be the collection of all ${\textstyle k}$ dimensional subspaces ${\textstyle W}$ such that ${\textstyle \operatorname {dim} \left(W\cap V_{j}\right)\geq j}$.

$$\lambda _{i_{1}}(A)+\cdots +\lambda _{i_{k}}(A)=\sup _{V_{1},\ldots ,V_{k}}\inf _{W\in X\left(V_{1},\ldots ,V_{k}\right)}tr_{W}(A)$$

Proof

Proof

The ${\textstyle \leq }$ case.

Let ${\textstyle V_{j}=span(e_{1},\dots ,e_{i_{j}})}$, and any ${\textstyle W\in X\left(V_{1},\ldots ,V_{k}\right)}$, it remains to show that $$\lambda _{i_{1}}(A)+\cdots +\lambda _{i_{k}}(A)\leq tr_{W}(A)$$

To show this, we construct an orthonormal set of vectors ${\textstyle v_{1},\dots ,v_{k}}$ such that ${\textstyle v_{j}\in V_{j}\cap W}$. Then ${\textstyle tr_{W}(A)\geq \sum _{j}\langle v_{j},Av_{j}\rangle \geq \lambda _{i_{j}}(A)}$

Since ${\textstyle dim(V_{1}\cap W)\geq 1}$, we pick any unit ${\textstyle v_{1}\in V_{1}\cap W}$. Next, since ${\textstyle dim(V_{2}\cap W)\geq 2}$, we pick any unit ${\textstyle v_{2}\in (V_{2}\cap W)}$ that is perpendicular to ${\textstyle v_{1}}$, and so on.

The ${\textstyle \geq }$ case.

For any such sequence of subspaces ${\textstyle V_{i}}$, we must find some ${\textstyle W\in X\left(V_{1},\ldots ,V_{k}\right)}$ such that $$\lambda _{i_{1}}(A)+\cdots +\lambda _{i_{k}}(A)\geq tr_{W}(A)$$

Now we prove this by induction.

The ${\textstyle n=1}$ case is the Courant-Fischer theorem. Assume now ${\textstyle n\geq 2}$.

If ${\textstyle i_{1}\geq 2}$, then we can apply induction. Let ${\textstyle E=span(e_{i_{1}},\dots ,e_{n})}$. We construct a partial flag within ${\textstyle E}$ from the intersection of ${\textstyle E}$ with ${\textstyle V_{1},\dots ,V_{k}}$.

We begin by picking a ${\textstyle (i_{k}-(i_{1}-1))}$-dimensional subspace ${\textstyle W_{k}'\subset E\cap V_{i_{k}}}$, which exists by counting dimensions. This has codimension ${\textstyle (i_{1}-1)}$ within ${\textstyle V_{i_{k}}}$.

Then we go down by one space, to pick a ${\textstyle (i_{k-1}-(i_{1}-1))}$-dimensional subspace ${\textstyle W_{k-1}'\subset W_{k}\cap V_{i_{k-1}}}$. This still exists. Etc. Now since ${\textstyle dim(E)\leq n-1}$, apply the induction hypothesis, there exists some ${\textstyle W\in X(W_{1},\dots ,W_{k})}$ such that $$\lambda _{i_{1}-(i_{1}-1)}(A|E)+\cdots +\lambda _{i_{k}-(i_{1}-1)}(A|E)\geq tr_{W}(A)$$ Now ${\textstyle \lambda _{i_{j}-(i_{1}-1)}(A|E)}$ is the ${\textstyle (i_{j}-(i_{1}-1))}$-th eigenvalue of ${\textstyle A}$ orthogonally projected down to ${\textstyle E}$. By Cauchy interlacing theorem, ${\textstyle \lambda _{i_{j}-(i_{1}-1)}(A|E)\leq \lambda _{i_{j}}(A)}$. Since ${\textstyle X(W_{1},\dots ,W_{k})\subset X(V_{1},\dots ,V_{k})}$, we’re done.

If ${\textstyle i_{1}=1}$, then we perform a similar construction. Let ${\textstyle E=span(e_{2},\dots ,e_{n})}$. If ${\textstyle V_{k}\subset E}$, then we can induct. Otherwise, we construct a partial flag sequence ${\textstyle W_{2},\dots ,W_{k}}$ By induction, there exists some ${\textstyle W'\in X(W_{2},\dots ,W_{k})\subset X(V_{2},\dots ,V_{k})}$, such that $$\lambda _{i_{2}-1}(A|E)+\cdots +\lambda _{i_{k}-1}(A|E)\geq tr_{W'}(A)$$ thus
$$\lambda _{i_{2}}(A)+\cdots +\lambda _{i_{k}}(A)\geq tr_{W'}(A)$$ And it remains to find some ${\textstyle v}$ such that ${\textstyle W'\oplus v\in X(V_{1},\dots ,V_{k})}$.

If ${\textstyle V_{1}\not \subset W'}$, then any ${\textstyle v\in V_{1}\setminus W'}$ would work. Otherwise, if ${\textstyle V_{2}\not \subset W'}$, then any ${\textstyle v\in V_{2}\setminus W'}$ would work, and so on. If none of these work, then it means ${\textstyle V_{k}\subset E}$, contradiction.

This have some corollaries:

Extremal partial trace — $$\lambda _{1}(A)+\dots +\lambda _{k}(A)=\sup _{\operatorname {dim} (V)=k}tr_{V}(A)$$

$$\xi _{1}(A)+\dots +\xi _{k}(A)=\inf _{\operatorname {dim} (V)=k}tr_{V}(A)$$

Corollary — The sum ${\textstyle \lambda _{1}(A)+\dots +\lambda _{k}(A)}$ is a convex function, and ${\textstyle \xi _{1}(A)+\dots +\xi _{k}(A)}$ is concave.

(Schur-Horn inequality) $$\xi _{1}(A)+\dots +\xi _{k}(A)\leq a_{i_{1},i_{1}}+\dots +a_{i_{k},i_{k}}\leq \lambda _{1}(A)+\dots +\lambda _{k}(A)$$ for any subset of indices.

Equivalently, this states that the diagonal vector of ${\textstyle A}$ is majorized by its eigenspectrum.

Schatten-norm Hölder inequality — Given Hermitian ${\textstyle A,B}$ and Hölder pair ${\textstyle 1/p+1/q=1}$, $$|\operatorname {tr} (AB)|\leq \|A\|_{S^{p}}\|B\|_{S^{q}}$$

Proof

Proof

WLOG, ${\textstyle B}$ is diagonalized, then we need to show ${\textstyle |\sum _{i}B_{ii}A_{ii}|\leq \|A\|_{S^{p}}\|(B_{ii})\|_{l^{q}}}$

By the standard Hölder inequality, it suffices to show ${\textstyle \|(A_{ii})\|_{l^{p}}\leq \|A\|_{S^{p}}}$

By the Schur-Horn inequality, the diagonals of ${\textstyle A}$ are majorized by the eigenspectrum of ${\textstyle A}$, and since the map ${\textstyle f(x_{1},\dots ,x_{n})=\|x\|_{p}}$ is symmetric and convex, it is Schur-convex.

Counterexample in the non-Hermitian case

Let N be the nilpotent matrix

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

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 ⁠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:

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

Similarly,

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

Here $\sigma _{k}^{\downarrow }$ denotes the k^th entry in the decreasing sequence of the singular values, so that $\sigma _{1}^{\downarrow }\geq \sigma _{2}^{\downarrow }\geq \cdots$.

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 PAP* = 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,

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

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

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

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

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

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.

Lidskii's inequality

Main article: Trace class § Lidskii's theorem

Lidskii inequality — If ${\textstyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ then $${\begin{aligned}&\lambda _{i_{1}}(A+B)+\cdots +\lambda _{i_{k}}(A+B)\\&\quad \leq \lambda _{i_{1}}(A)+\cdots +\lambda _{i_{k}}(A)+\lambda _{1}(B)+\cdots +\lambda _{k}(B)\end{aligned}}$$

$${\begin{aligned}&\lambda _{i_{1}}(A+B)+\cdots +\lambda _{i_{k}}(A+B)\\&\quad \geq \lambda _{i_{1}}(A)+\cdots +\lambda _{i_{k}}(A)+\xi _{1}(B)+\cdots +\xi _{k}(B)\end{aligned}}$$

Proof

Proof

The second is the negative of the first. The first is by Wielandt minimax.

$${\begin{aligned}&\lambda _{i_{1}}(A+B)+\cdots +\lambda _{i_{k}}(A+B)\\=&\sup _{V_{1},\dots ,V_{k}}\inf _{W\in X(V_{1},\dots ,V_{k})}(tr_{W}(A)+tr_{W}(B))\\=&\sup _{V_{1},\dots ,V_{k}}(\inf _{W\in X(V_{1},\dots ,V_{k})}tr_{W}(A)+tr_{W}(B))\\\leq &\sup _{V_{1},\dots ,V_{k}}(\inf _{W\in X(V_{1},\dots ,V_{k})}tr_{W}(A)+(\lambda _{1}(B)+\cdots +\lambda _{k}(B)))\\=&\lambda _{i_{1}}(A)+\cdots +\lambda _{i_{k}}(A)+\lambda _{1}(B)+\cdots +\lambda _{k}(B)\end{aligned}}$$

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

p-Wielandt-Hoffman inequality — ${\textstyle \|\lambda (A+B)-\lambda (A)\|_{\ell ^{p}}\leq \|B\|_{S^{p}}}$ where ${\textstyle \|\cdot \|_{S^{p}}}$ stands for the p-Schatten norm.

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

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

where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write $\lambda _{k}=\lambda _{k}^{\downarrow }$.) 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:

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

A similar pair of equalities hold for negative eigenvalues.

Proof

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

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

Therefore, for all S_k

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

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:

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

So

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

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

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

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},

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

So

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

This implies

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

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

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

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

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

Self-adjoint operators

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

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

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

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

If we only have N eigenvalues and hence run out of eigenvalues, then we let $E_{n}:=\inf \sigma _{ess}(A)$ (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 $(A-E)\geq 0$ for $E\in \mathbb {R}$ if and only if $\sigma (A)\subseteq [E,\infty )$.

Theorem. If A is self-adjoint, then

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

and

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

See also

• Courant minimax principle
• Max–min inequality

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.
• 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.