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.