Bauer–Fike theorem

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

In Mathematik, 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. T. Fike in 1960.

Inhalt 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: {Anzeigestil kappa _{p}(X)=|X|_{p}links|X^{-1}Rechts|_{p}.} The Bauer–Fike Theorem Bauer–Fike Theorem. Let μ be an eigenvalue of A + δA. Then there exists λ ∈ Λ(EIN) so dass: {Anzeigestil |lambda -mu |leq kappa _{p}(v)|delta A|_{p}} Nachweisen. We can suppose μ ∉ Λ(EIN), 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(EIN + δA − μI) = 0 und so {Anzeigestil {Start{ausgerichtet}0&=det(A+delta A-mu I)\&=det(V^{-1})das(A+delta A-mu I)das(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)es ging((Lambda -mu I)^{-1}V^{-1}delta AV+Iright)\Ende{ausgerichtet}}} However our assumption, μ ∉ Λ(EIN), impliziert, dass: das(Λ − μI) 0 and therefore we can write: {displaystyle det left((Lambda -mu I)^{-1}V^{-1}delta AV+Iright)=0.} This reveals −1 to be an eigenvalue of {Anzeigestil (Lambda -mu I)^{-1}V^{-1}delta AV.} Since all p-norms are consistent matrix norms we have |l| ≤ ||EIN||p where λ is an eigenvalue of A. In this instance this gives us: {displaystyle 1=|-1|leq left|(Lambda -mu I)^{-1}V^{-1}delta AVright|_{p}leq left|(Lambda -mu I)^{-1}Rechts|_{p}links|V^{-1}Rechts|_{p}|v|_{p}|delta A|_{p}=links|(Lambda -mu I)^{-1}Rechts|_{p} kappa _{p}(v)|delta A|_{p}} Aber (Λ − μI)−1 is a diagonal matrix, the p-norm of which is easily computed: {Anzeigestil links|links(Lambda -mu Iright)^{-1}Rechts|_{p} = max _{|{Fettsymbol {x}}|_{p}neq 0}{frac {links|links(Lambda -mu Iright)^{-1}{Fettsymbol {x}}Rechts|_{p}}{|{Fettsymbol {x}}|_{p}}}= max _{Lambda in Lambda (EIN)}{frac {1}{|lambda -mu |}} ={frac {1}{Mindest _{Lambda in Lambda (EIN)}|lambda -mu |}}} whence: {Anzeigestil min _{Lambda in Lambda (EIN)}|lambda -mu |leq kappa _{p}(v)|delta A|_{p}.} An Alternate Formulation The theorem can also be reformulated to better suit numerical methods. In der Tat, 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). Lassen (λa, va ) be an approximate eigenvalue-eigenvector couple, and r = Ava − λava. Then there exists λ ∈ Λ(EIN) so dass: {Anzeigestil links|lambda -lambda ^{a}Rechts|leq kappa _{p}(v){frac {|{Fettsymbol {r}}|_{p}}{links|{Fettsymbol {v}}^{a}Rechts|_{p}}}} Nachweisen. We can suppose λa ∉ Λ(EIN), otherwise take λ = λa and the result is trivially true since κp(v) ≥ 1. So (A − λaI)−1 exists, so we can write: {Anzeigestil {Fettsymbol {v}}^{a}=links(A-lambda ^{a}Iright)^{-1}{Fettsymbol {r}}=Vleft(D-lambda ^{a}Iright)^{-1}V^{-1}{Fettsymbol {r}}} since A is diagonalizable; taking the p-norm of both sides, wir erhalten: {Anzeigestil links|{Fettsymbol {v}}^{a}Rechts|_{p}=links|Gültigkeit(D-lambda ^{a}Iright)^{-1}V^{-1}{Fettsymbol {r}}Rechts|_{p}leq |v|_{p}links|links(D-lambda ^{a}Iright)^{-1}Rechts|_{p}links|V^{-1}Rechts|_{p}|{Fettsymbol {r}}|_{p}=kappa _{p}(v)links|links(D-lambda ^{a}Iright)^{-1}Rechts|_{p}|{Fettsymbol {r}}|_{p}.} Jedoch {Anzeigestil links(D-lambda ^{a}Iright)^{-1}} is a diagonal matrix and its p-norm is easily computed: {Anzeigestil links|links(D-lambda ^{a}Iright)^{-1}Rechts|_{p}= max _{|{Fettsymbol {x}}|_{p}neq 0}{frac {links|links(D-lambda ^{a}Iright)^{-1}{Fettsymbol {x}}Rechts|_{p}}{|{Fettsymbol {x}}|_{p}}}= max _{Lambda im Sigma (EIN)}{frac {1}{links|lambda -lambda ^{a}Rechts|}}={frac {1}{Mindest _{Lambda im Sigma (EIN)}links|lambda -lambda ^{a}Rechts|}}} whence: {Anzeigestil min _{lambda in lambda (EIN)}links|lambda -lambda ^{a}Rechts|leq kappa _{p}(v){frac {|{Fettsymbol {r}}|_{p}}{links|{Fettsymbol {v}}^{a}Rechts|_{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: Logische Folge. Suppose A is invertible and that μ is an eigenvalue of A + δA. Then there exists λ ∈ Λ(EIN) so dass: {Anzeigestil {frac {|lambda -mu |}{|Lambda |}}leq kappa _{p}(v)links|A^{-1}delta Aright|_{p}} Notiz. ||A−1δA|| can be formally viewed as the relative variation of A, just as |λ − μ| / |l| is the relative variation of λ.

Nachweisen. Since μ is an eigenvalue of A + δA and det(EIN) 0, by multiplying by −A−1 from left we have: {displaystyle -A^{-1}(A+delta A){Fettsymbol {v}}=-mu A^{-1}{Fettsymbol {v}}.} If we set: {Anzeigestil A^{a}=mu A^{-1},Quad (delta A)^{a}=-A^{-1}delta A} then we have: {Anzeigestil links(A^{a}+(delta A)^{a}-Iright){Fettsymbol {v}}={Fettsymbol {0}}} which means that 1 is an eigenvalue of Aa + (δA)a, with v as an eigenvector. Jetzt, the eigenvalues of Aa are μ / λi , while it has the same eigenvector matrix as A. Applying the Bauer–Fike theorem to Aa + (δA)a with eigenvalue 1, gives us: {Anzeigestil min _{Lambda in Lambda (EIN)}links|{frac {in }{Lambda }}-1Rechts|= min _{Lambda in Lambda (EIN)}{frac {|lambda -mu |}{|Lambda |}}leq kappa _{p}(v)links|A^{-1}delta Aright|_{p}} The Case of Normal Matrices If A is normal, V is a unitary matrix, deshalb: {Anzeigestil |v|_{2}=links|V^{-1}Rechts|_{2}=1,} so that κ2(v) = 1. The Bauer–Fike theorem then becomes: {displaystyle exists lambda in Lambda (EIN):Quad |lambda -mu |leq |delta A|_{2}} Or in alternate formulation: {displaystyle exists lambda in Lambda (EIN):quad left|lambda -lambda ^{a}Rechts|leq {frac {|{Fettsymbol {r}}|_{2}}{links|{Fettsymbol {v}}^{a}Rechts|_{2}}}} which obviously remains true if A is a Hermitian matrix. In diesem Fall, jedoch, 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(EIN) 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. T. (1960). "Norms and Exclusion Theorems". Numer. Mathematik. 2 (1): 137–141. doi:10.1007/BF01386217. S2CID 121278235. Eisenstat, S. C.; Ipsen, ich. C. F. (1998). "Three absolute perturbation bounds for matrix eigenvalues imply relative bounds". SIAM Journal über Matrixanalyse und Anwendungen. 20 (1): 149–158. CiteSeerX 10.1.1.45.3999. doi:10.1137/S0895479897323282. verbergen vte Funktionsanalyse (Themen – Glossar) Leerzeichen BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraisch/topologisch)lokal konvexreflexivseparable Theoreme Hahn-BanachRiesz-Darstellunggeschlossener Graphgleichmäßiges BeschränktheitsprinzipKakutani-FixpunktKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatoren adjointboundcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebren Banach-AlgebraC*-AlgebraSpektrum einer C*-AlgebraOperator-Algebravon Gruppenalgebra einer lokalvariant-kompakten Gruppe SubraumproblemMahlersche Vermutung Anwendungen Hardy-RaumSpektraltheorie gewöhnlicher DifferentialgleichungenWärmekernindexsatzVariationsrechnungFunktionsrechnungIntegraloperatorJones-PolynomTopologische QuantenfeldtheorieNichtkommutative GeometrieRiemann-HypotheseVerteilung (oder verallgemeinerte Funktionen) Fortgeschrittene Themen Approximation PropertyBalanced SetChoquet-TheorieSchwache TopologieBanach-Mazur-AbstandTomita-Takesaki-Theorie Kategorien: Spectral theoryTheorems in analysis

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