Min-max theorem

Min-max theorem Not to be confused with Minimax theorem. "Variational theorem" redirects here. Not to be confused with variational principle. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Min-max theorem" – news · newspapers · books · scholar · JSTOR (November 2011) (Learn how and when to remove this template message) 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.

Contents 1 Matrices 1.1 Min-max theorem 1.2 Counterexample in the non-Hermitian case 2 Applications 2.1 Min-max principle for singular values 2.2 Cauchy interlacing theorem 3 Compact operators 4 Self-adjoint operators 5 See also 6 References 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 {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 RA(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 ≤ ... ≤ λk ≤ ... ≤ λn then {displaystyle lambda _{k}=min _{U}{max _{x}{R_{A}(x)mid xin U{text{ and }}xneq 0}mid dim(U)=k}} and {displaystyle lambda _{k}=max _{U}{min _{x}{R_{A}(x)mid xin U{text{ and }}xneq 0}mid dim(U)=n-k+1}} in particular, {displaystyle lambda _{1}leq R_{A}(x)leq lambda _{n}quad forall xin 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 λn is given by: {displaystyle lambda _{n}=max{R_{A}(x):xneq 0}.} Similarly, the minimal eigenvalue λ1 is given by: {displaystyle lambda _{1}=min{R_{A}(x):xneq 0}.} Proof Since the matrix A is Hermitian it is diagonalizable and we can choose an orthonormal basis of eigenvectors {u1, ..., un} that is, ui is an eigenvector for the eigenvalue λi and such that (ui, ui) = 1 and (ui, uj) = 0 for all i ≠ j.

If U is a subspace of dimension k then its intersection with the subspace span{uk, ..., un} isn't zero, for if it were, then the dimension of the span of the two subspaces would be {displaystyle k+(n-k+1)} , which is impossible. 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 xin U}geq lambda _{k}} Since this is true for all U, we can conclude that {displaystyle min{max{R_{A}(x)mid xin U{text{ and }}xneq 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, ..., uk} , for which {displaystyle max{R_{A}(x)mid xin V{text{ and }}xneq 0}leq lambda _{k}} because {displaystyle lambda _{k}} is the largest eigenvalue in V. Therefore, also {displaystyle min{max{R_{A}(x)mid xin U{text{ and }}xneq 0}mid dim(U)=k}leq lambda _{k}} To get the other formula, consider the Hermitian matrix {displaystyle A'=-A} , whose eigenvalues in increasing order are {displaystyle lambda '_{k}=-lambda _{n-k+1}} . Applying the result just proved, {displaystyle {begin{aligned}-lambda _{n-k+1}=lambda '_{k}&=min{max{R_{A'}(x)mid xin U}mid dim(U)=k}\&=min{max{-R_{A}(x)mid xin U}mid dim(U)=k}\&=-max{min{R_{A}(x)mid xin U}mid dim(U)=k}end{aligned}}} The result follows on replacing {displaystyle k} with {displaystyle n-k+1} .

Counterexample in the non-Hermitian case Let N be the nilpotent matrix {displaystyle {begin{bmatrix}0&1\0&0end{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 {σk} of a square matrix M are the square roots of the eigenvalues of M*M (equivalently MM*). An immediate consequence[citation needed] of the first equality in the min-max theorem is: {displaystyle sigma _{k}^{uparrow }=min _{S:dim(S)=k}max _{xin S,|x|=1}(M^{*}Mx,x)^{frac {1}{2}}=min _{S:dim(S)=k}max _{xin S,|x|=1}|Mx|.} Similarly, {displaystyle sigma _{k}^{uparrow }=max _{S:dim(S)=n-k+1}min _{xin S,|x|=1}|Mx|.} Here {displaystyle sigma _{k}=sigma _{k}^{uparrow }} denotes the kth 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 PAP* = B. The Cauchy interlacing theorem states: Theorem. If the eigenvalues of A are α1 ≤ ... ≤ αn, and those of B are β1 ≤ ... ≤ βj ≤ ... ≤ βm, 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 βi have corresponding eigenvector bi and Sj be the j dimensional subspace Sj = span{b1, ..., bj}, then {displaystyle beta _{j}=max _{xin S_{j},|x|=1}(Bx,x)=max _{xin S_{j},|x|=1}(PAP^{*}x,x)geq min _{S_{j}}max _{xin S_{j},|x|=1}(A(P^{*}x),P^{*}x)=alpha _{j}.} According to first part of min-max, αj ≤ βj. On the other hand, if we define Sm−j+1 = span{bj, ..., bm}, then {displaystyle beta _{j}=min _{xin S_{m-j+1},|x|=1}(Bx,x)=min _{xin S_{m-j+1},|x|=1}(PAP^{*}x,x)=min _{xin 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 αj ≤ βj ≤ αj+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 Sk ⊂ 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 ... ≤ λk ≤ ... ≤ λ1. Then: {displaystyle {begin{aligned}max _{S_{k}}min _{xin S_{k},|x|=1}(Ax,x)&=lambda _{k}^{downarrow },\min _{S_{k-1}}max _{xin 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' ∩ Sk is non empty. So there exists x ∈ S'  ∩ Sk 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 Sk {displaystyle inf _{xin 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 _{xin S_{k},|x|=1}(Ax,x)leq lambda _{k}.} So {displaystyle sup _{S_{k}}min _{xin 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 _{xin 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 Sk−1, whose the orthogonal complement is denoted by Sk−1⊥. If S' = span{u1...uk}, {displaystyle S'cap S_{k-1}^{perp }neq {0}.} So {displaystyle exists xin S_{k-1}^{perp },|x|=1,(Ax,x)geq lambda _{k}.} This implies {displaystyle max _{xin 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 _{xin S_{k-1}^{perp },|x|=1}(Ax,x)geq lambda _{k}.} Pick Sk−1 = span{u1, ..., uk−1} and we deduce {displaystyle min _{S_{k-1}}max _{xin 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 ,Apsi 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 ,Apsi 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 Ein 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 ,Apsi rangle } and {displaystyle sup sigma (A)=sup _{psi in {mathfrak {D}}(A),|psi |=1}langle psi ,Apsi rangle } .[1]: 77  See also Courant minimax principle Max–min inequality References ^ Jump up to: a b c d G. Teschl, Mathematical Methods in Quantum Mechanics (GSM 99) https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf ^ Jump up to: a b Lieb; Loss (2001). Analysis. GSM. Vol. 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. show vte Functional analysis (topics – glossary) show vte Analysis in topological vector spaces show vte Spectral theory and *-algebras Categories: Theorems in functional analysisSpectral theoryOperator theory

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