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. 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.
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| - Satz von Courant-Fischer (de)
- Théorème min-max de Courant-Fischer (fr)
- Min-max theorem (en)
- Теорема Куранта — Фишера (ru)
- Теорема Куранта — Фішера (uk)
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| - Der Satz von Courant-Fischer (auch Minimum-Maximum-Prinzip) ist ein mathematischer Satz aus der linearen Algebra, der eine variationelle Charakterisierung der Eigenwerte einer symmetrischen oder hermiteschen Matrix ermöglicht. Jeder Eigenwert wird dabei als minimaler beziehungsweise maximaler Rayleigh-Quotient von Vektoren aus Untervektorräumen mit bestimmten Dimensionen dargestellt. Der Satz ist nach den Mathematikern Richard Courant und Ernst Fischer benannt. Er dient unter anderem zur Eigenwertabschätzung und zur Analyse numerischer Eigenwertverfahren. (de)
- En algèbre linéaire et en analyse fonctionnelle, le théorème min-max de Courant-Fischer donne une caractérisation variationnelle des valeurs propres d'une matrice hermitienne. Il permet donc de caractériser les valeurs singulières d'une matrice complexe quelconque. Il s'étend aux opérateurs compacts autoadjoints sur un espace de Hilbert, ainsi qu'aux opérateurs autoadjoints bornés inférieurement. (fr)
- Теорема Куранта — Фишера — теорема о свойстве эрмитова оператора в гильбертовом пространстве функций. Также называется теоремой о минимаксе. (ru)
- Теорема Куранта — Фішера — теорема про властивість ермітового оператора в гільбертовому просторі функцій. Також називається теоремою про мінімакс. (uk)
- 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. 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. (en)
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| proof
| - Since the matrix 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 = 1 and = 0 for all i ≠ j.
If U is a subspace of dimension k then its intersection with the subspace isn't zero,
for if it were, then the dimension of the span of the two subspaces would be , which is impossible. Hence there exists a vector in this intersection that we can write as
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and whose Rayleigh quotient is
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and hence
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Since this is true for all U, we can conclude that
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This is one inequality. To establish the other inequality, chose the specific k-dimensional space
, for which
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because is the largest eigenvalue in V. Therefore, also
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To get the other formula, consider the Hermitian matrix , whose eigenvalues in increasing order are .
Applying the result just proved,
:
The result follows on replacing with . (en)
- Let S' be the closure of the linear span .
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 . Since it is an element of S' , such an x necessarily satisfy
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Therefore, for all Sk
:
But is compact, therefore the function f = is weakly continuous. Furthermore, any bounded set in H is weakly compact. This lets us replace the infimum by minimum:
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So
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Because equality is achieved when ,
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This is the first part of min-max theorem for compact self-adjoint operators.
Analogously, consider now a -dimensional subspace S'k−1, whose the orthogonal complement is denoted by S'k−1⊥. If S' = span{u1...uk},
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So
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This implies
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where the compactness of A was applied. Index the above by the collection of k-1-dimensional subspaces gives
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Pick S'k−1 = span{u1, ..., u'k−1} and we deduce
: (en)
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| - Der Satz von Courant-Fischer (auch Minimum-Maximum-Prinzip) ist ein mathematischer Satz aus der linearen Algebra, der eine variationelle Charakterisierung der Eigenwerte einer symmetrischen oder hermiteschen Matrix ermöglicht. Jeder Eigenwert wird dabei als minimaler beziehungsweise maximaler Rayleigh-Quotient von Vektoren aus Untervektorräumen mit bestimmten Dimensionen dargestellt. Der Satz ist nach den Mathematikern Richard Courant und Ernst Fischer benannt. Er dient unter anderem zur Eigenwertabschätzung und zur Analyse numerischer Eigenwertverfahren. (de)
- En algèbre linéaire et en analyse fonctionnelle, le théorème min-max de Courant-Fischer donne une caractérisation variationnelle des valeurs propres d'une matrice hermitienne. Il permet donc de caractériser les valeurs singulières d'une matrice complexe quelconque. Il s'étend aux opérateurs compacts autoadjoints sur un espace de Hilbert, ainsi qu'aux opérateurs autoadjoints bornés inférieurement. (fr)
- 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. (en)
- Теорема Куранта — Фишера — теорема о свойстве эрмитова оператора в гильбертовом пространстве функций. Также называется теоремой о минимаксе. (ru)
- Теорема Куранта — Фішера — теорема про властивість ермітового оператора в гільбертовому просторі функцій. Також називається теоремою про мінімакс. (uk)
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