Bauer–Fike theorem

Bauer–Fike theorem For the theorem in algebraic number theory, see Bauer's theorem.

En mathématiques, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally speaking, what it says is that the sensitivity of the eigenvalues is estimated by the condition number of the matrix of eigenvectors.

The theorem was proved by Friedrich L. Bauer and C. J. Fike in 1960.

Contenu 1 The setup 2 The Bauer–Fike Theorem 3 An Alternate Formulation 4 A Relative Bound 5 The Case of Normal Matrices 6 References The setup In what follows we assume that: A ∈ Cn,n is a diagonalizable matrix; V ∈ Cn,n is the non-singular eigenvector matrix such that A = VΛV −1, where Λ is a diagonal matrix. If X ∈ Cn,n is invertible, its condition number in p-norm is denoted by κp(X) and defined by: {displaystyle kappa _{p}(X)=|X|_{p}la gauche|X^{-1}droit|_{p}.} The Bauer–Fike Theorem Bauer–Fike Theorem. Let μ be an eigenvalue of A + δA. Then there exists λ ∈ Λ(UN) tel que: {style d'affichage |lambda -mu |leq kappa _{p}(V)|delta A|_{p}} Preuve. We can suppose μ ∉ Λ(UN), otherwise take λ = μ and the result is trivially true since κp(V) ≥ 1. Since μ is an eigenvalue of A + δA, we have det(UN + δA − μI) = 0 et donc {style d'affichage {commencer{aligné}0&=det(A+delta A-mu I)\&=det(V^{-1})la(A+delta A-mu I)la(V)\&=det left(V^{-1}(A+delta A-mu I)Vright)\&=det left(V^{-1}AV+V^{-1}delta AV-V^{-1}mu IVright)\&=det left(Lambda +V^{-1}delta AV-mu Iright)\&=det(Lambda -mu I)il est parti((Lambda -mu I)^{-1}V^{-1}delta AV+Iright)\fin{aligné}}} However our assumption, μ ∉ Λ(UN), implique que: la(Λ − μI) ≠ 0 and therefore we can write: {style d'affichage à gauche((Lambda -mu I)^{-1}V^{-1}delta AV+Iright)=0.} This reveals −1 to be an eigenvalue of {style d'affichage (Lambda -mu I)^{-1}V^{-1}delta AV.} Since all p-norms are consistent matrix norms we have |je| ≤ ||UN||p where λ is an eigenvalue of A. In this instance this gives us: {displaystyle 1=|-1|gauche|(Lambda -mu I)^{-1}V^{-1}delta AVright|_{p}gauche|(Lambda -mu I)^{-1}droit|_{p}la gauche|V^{-1}droit|_{p}|V|_{p}|delta A|_{p}=gauche|(Lambda -mu I)^{-1}droit|_{p} kappa _{p}(V)|delta A|_{p}} Mais (Λ − μI)−1 is a diagonal matrix, the p-norm of which is easily computed: {style d'affichage à gauche|la gauche(Lambda -mu Iright)^{-1}droit|_{p} =max _{|{symbole gras {X}}|_{p}neq 0}{frac {la gauche|la gauche(Lambda -mu Iright)^{-1}{symbole gras {X}}droit|_{p}}{|{symbole gras {X}}|_{p}}}=max _{lambda dans lambda (UN)}{frac {1}{|lambda -mu |}} ={frac {1}{min _{lambda dans lambda (UN)}|lambda -mu |}}} whence: {style d'affichage min _{lambda dans lambda (UN)}|lambda -mu |leq kappa _{p}(V)|delta A|_{p}.} An Alternate Formulation The theorem can also be reformulated to better suit numerical methods. En réalité, dealing with real eigensystem problems, one often has an exact matrix A, but knows only an approximate eigenvalue-eigenvector couple, (λa, va ) and needs to bound the error. The following version comes in help.

Bauer–Fike Theorem (Alternate Formulation). Laisser (λa, va ) be an approximate eigenvalue-eigenvector couple, and r = Ava − λava. Then there exists λ ∈ Λ(UN) tel que: {style d'affichage à gauche|lambda -lambda ^{un}droit|leq kappa _{p}(V){frac {|{symbole gras {r}}|_{p}}{la gauche|{symbole gras {v}}^{un}droit|_{p}}}} Preuve. We can suppose λa ∉ Λ(UN), otherwise take λ = λa and the result is trivially true since κp(V) ≥ 1. Alors (A − λaI)−1 exists, so we can write: {style d'affichage {symbole gras {v}}^{un}=gauche(A-lambda ^{un}Iright)^{-1}{symbole gras {r}}=Vleft(D-lambda ^{un}Iright)^{-1}V^{-1}{symbole gras {r}}} since A is diagonalizable; taking the p-norm of both sides, on obtient: {style d'affichage à gauche|{symbole gras {v}}^{un}droit|_{p}=gauche|Vleft(D-lambda ^{un}Iright)^{-1}V^{-1}{symbole gras {r}}droit|_{p}leq |V|_{p}la gauche|la gauche(D-lambda ^{un}Iright)^{-1}droit|_{p}la gauche|V^{-1}droit|_{p}|{symbole gras {r}}|_{p}=kappa _{p}(V)la gauche|la gauche(D-lambda ^{un}Iright)^{-1}droit|_{p}|{symbole gras {r}}|_{p}.} Cependant {style d'affichage à gauche(D-lambda ^{un}Iright)^{-1}} is a diagonal matrix and its p-norm is easily computed: {style d'affichage à gauche|la gauche(D-lambda ^{un}Iright)^{-1}droit|_{p}=max _{|{symbole gras {X}}|_{p}neq 0}{frac {la gauche|la gauche(D-lambda ^{un}Iright)^{-1}{symbole gras {X}}droit|_{p}}{|{symbole gras {X}}|_{p}}}=max _{lambda en sigma (UN)}{frac {1}{la gauche|lambda -lambda ^{un}droit|}}={frac {1}{min _{lambda en sigma (UN)}la gauche|lambda -lambda ^{un}droit|}}} whence: {style d'affichage min _{lambda in lambda (UN)}la gauche|lambda -lambda ^{un}droit|leq kappa _{p}(V){frac {|{symbole gras {r}}|_{p}}{la gauche|{symbole gras {v}}^{un}droit|_{p}}}.} A Relative Bound Both formulations of Bauer–Fike theorem yield an absolute bound. The following corollary is useful whenever a relative bound is needed: Corollaire. Suppose A is invertible and that μ is an eigenvalue of A + δA. Then there exists λ ∈ Λ(UN) tel que: {style d'affichage {frac {|lambda -mu |}{|lambda |}}leq kappa _{p}(V)la gauche|Un ^{-1}delta Aright|_{p}} Noter. ||A−1δA|| can be formally viewed as the relative variation of A, just as |λ − μ| / |je| is the relative variation of λ.

Preuve. Since μ is an eigenvalue of A + δA and det(UN) ≠ 0, by multiplying by −A−1 from left we have: {displaystyle -A^{-1}(A+delta A){symbole gras {v}}=-mu A^{-1}{symbole gras {v}}.} If we set: {style d'affichage A^{un}=mu A^{-1},qquad (delta A)^{un}=-A^{-1}delta A} then we have: {style d'affichage à gauche(Un ^{un}+(delta A)^{un}-Iright){symbole gras {v}}={symbole gras {0}}} which means that 1 is an eigenvalue of Aa + (δA)un, with v as an eigenvector. À présent, the eigenvalues of Aa are μ / je , while it has the same eigenvector matrix as A. Applying the Bauer–Fike theorem to Aa + (δA)a with eigenvalue 1, gives us: {style d'affichage min _{lambda dans lambda (UN)}la gauche|{frac {dans }{lambda }}-1droit|=min _{lambda dans lambda (UN)}{frac {|lambda -mu |}{|lambda |}}leq kappa _{p}(V)la gauche|Un ^{-1}delta Aright|_{p}} The Case of Normal Matrices If A is normal, V is a unitary matrix, Donc: {style d'affichage |V|_{2}=gauche|V^{-1}droit|_{2}=1,} so that κ2(V) = 1. The Bauer–Fike theorem then becomes: {displaystyle exists lambda in Lambda (UN):quad |lambda -mu |leq |delta A|_{2}} Or in alternate formulation: {displaystyle exists lambda in Lambda (UN):quad left|lambda -lambda ^{un}droit|leq {frac {|{symbole gras {r}}|_{2}}{la gauche|{symbole gras {v}}^{un}droit|_{2}}}} which obviously remains true if A is a Hermitian matrix. Dans ce cas, toutefois, a much stronger result holds, known as the Weyl's theorem on eigenvalues. In the hermitian case one can also restate the Bauer–Fike theorem in the form that the map A ↦ Λ(UN) that maps a matrix to its spectrum is a non-expansive function with respect to the Hausdorff distance on the set of compact subsets of C.

References Bauer, F. L; Fike, C. J. (1960). "Norms and Exclusion Theorems". Numer. Math. 2 (1): 137–141. est ce que je:10.1007/BF01386217. S2CID 121278235. Eisenstat, S. C; Ipsen, je. C. F. (1998). "Three absolute perturbation bounds for matrix eigenvalues imply relative bounds". SIAM Journal sur l'analyse matricielle et les applications. 20 (1): 149–158. CiteSeerX 10.1.1.45.3999. est ce que je:10.1137/S0895479897323282. cacher vte Analyse fonctionnelle (sujets – glossaire) Espaces BanachBesovFréchetHilbertHölderNucléaireOrliczSchwartzSobolevvecteur topologique Propriétés tonneaucomplètedouble (algébrique/topologique)localement convexe réflexif séparable Théorèmes Hahn–Banach Représentation de Riesz graphe fermé principe de délimitation uniforme Kakutani virgule fixeKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Opérateurs adjointlimitécompactHilbert–Schmidtnormalnucléairetraceclasstransposéillimitéunitaire problème de sous-espaceconjecture de MahlerApplicationsespace de Hardythéorie spectrale des équations différentielles ordinairesnoyau de chaleurthéorème d'indexcalcul des variationscalcul fonctionnelopérateur intégralpolynôme de Jonesthéorie des champs quantiques topologiquesgéométrie non commutativehypothèse de Riemanndistribution (ou fonctions généralisées) Sujets avancés propriété d'approximationensemble équilibréThéorie de Choquettopologie faibleDistance de Banach–MazurThéorie de Tomita–Takesaki Catégories: Spectral theoryTheorems in analysis