Bauer–Fike theorem

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

In mathematics, 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.

Contents 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}left|X^{-1}right|_{p}.} The Bauer–Fike Theorem Bauer–Fike Theorem. Let μ be an eigenvalue of A + δA. Then there exists λ ∈ Λ(A) such that: {displaystyle |lambda -mu |leq kappa _{p}(V)|delta A|_{p}} Proof. We can suppose μ ∉ Λ(A), 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(A + δA − μI) = 0 and so {displaystyle {begin{aligned}0&=det(A+delta A-mu I)\&=det(V^{-1})det(A+delta A-mu I)det(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)det left((Lambda -mu I)^{-1}V^{-1}delta AV+Iright)\end{aligned}}} However our assumption, μ ∉ Λ(A), implies that: det(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 {displaystyle (Lambda -mu I)^{-1}V^{-1}delta AV.} Since all p-norms are consistent matrix norms we have |λ| ≤ ||A||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}right|_{p}left|V^{-1}right|_{p}|V|_{p}|delta A|_{p}=left|(Lambda -mu I)^{-1}right|_{p} kappa _{p}(V)|delta A|_{p}} But (Λ − μI)−1 is a diagonal matrix, the p-norm of which is easily computed: {displaystyle left|left(Lambda -mu Iright)^{-1}right|_{p} =max _{|{boldsymbol {x}}|_{p}neq 0}{frac {left|left(Lambda -mu Iright)^{-1}{boldsymbol {x}}right|_{p}}{|{boldsymbol {x}}|_{p}}}=max _{lambda in Lambda (A)}{frac {1}{|lambda -mu |}} ={frac {1}{min _{lambda in Lambda (A)}|lambda -mu |}}} whence: {displaystyle min _{lambda in Lambda (A)}|lambda -mu |leq kappa _{p}(V)|delta A|_{p}.} An Alternate Formulation The theorem can also be reformulated to better suit numerical methods. In fact, 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). Let (λa, va ) be an approximate eigenvalue-eigenvector couple, and r = Ava − λava. Then there exists λ ∈ Λ(A) such that: {displaystyle left|lambda -lambda ^{a}right|leq kappa _{p}(V){frac {|{boldsymbol {r}}|_{p}}{left|{boldsymbol {v}}^{a}right|_{p}}}} Proof. We can suppose λa ∉ Λ(A), otherwise take λ = λa and the result is trivially true since κp(V) ≥ 1. So (A − λaI)−1 exists, so we can write: {displaystyle {boldsymbol {v}}^{a}=left(A-lambda ^{a}Iright)^{-1}{boldsymbol {r}}=Vleft(D-lambda ^{a}Iright)^{-1}V^{-1}{boldsymbol {r}}} since A is diagonalizable; taking the p-norm of both sides, we obtain: {displaystyle left|{boldsymbol {v}}^{a}right|_{p}=left|Vleft(D-lambda ^{a}Iright)^{-1}V^{-1}{boldsymbol {r}}right|_{p}leq |V|_{p}left|left(D-lambda ^{a}Iright)^{-1}right|_{p}left|V^{-1}right|_{p}|{boldsymbol {r}}|_{p}=kappa _{p}(V)left|left(D-lambda ^{a}Iright)^{-1}right|_{p}|{boldsymbol {r}}|_{p}.} However {displaystyle left(D-lambda ^{a}Iright)^{-1}} is a diagonal matrix and its p-norm is easily computed: {displaystyle left|left(D-lambda ^{a}Iright)^{-1}right|_{p}=max _{|{boldsymbol {x}}|_{p}neq 0}{frac {left|left(D-lambda ^{a}Iright)^{-1}{boldsymbol {x}}right|_{p}}{|{boldsymbol {x}}|_{p}}}=max _{lambda in sigma (A)}{frac {1}{left|lambda -lambda ^{a}right|}}={frac {1}{min _{lambda in sigma (A)}left|lambda -lambda ^{a}right|}}} whence: {displaystyle min _{lambda in lambda (A)}left|lambda -lambda ^{a}right|leq kappa _{p}(V){frac {|{boldsymbol {r}}|_{p}}{left|{boldsymbol {v}}^{a}right|_{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: Corollary. Suppose A is invertible and that μ is an eigenvalue of A + δA. Then there exists λ ∈ Λ(A) such that: {displaystyle {frac {|lambda -mu |}{|lambda |}}leq kappa _{p}(V)left|A^{-1}delta Aright|_{p}} Note. ||A−1δA|| can be formally viewed as the relative variation of A, just as |λ − μ| / |λ| is the relative variation of λ.

Proof. Since μ is an eigenvalue of A + δA and det(A) ≠ 0, by multiplying by −A−1 from left we have: {displaystyle -A^{-1}(A+delta A){boldsymbol {v}}=-mu A^{-1}{boldsymbol {v}}.} If we set: {displaystyle A^{a}=mu A^{-1},qquad (delta A)^{a}=-A^{-1}delta A} then we have: {displaystyle left(A^{a}+(delta A)^{a}-Iright){boldsymbol {v}}={boldsymbol {0}}} which means that 1 is an eigenvalue of Aa + (δA)a, with v as an eigenvector. Now, 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: {displaystyle min _{lambda in Lambda (A)}left|{frac {mu }{lambda }}-1right|=min _{lambda in Lambda (A)}{frac {|lambda -mu |}{|lambda |}}leq kappa _{p}(V)left|A^{-1}delta Aright|_{p}} The Case of Normal Matrices If A is normal, V is a unitary matrix, therefore: {displaystyle |V|_{2}=left|V^{-1}right|_{2}=1,} so that κ2(V) = 1. The Bauer–Fike theorem then becomes: {displaystyle exists lambda in Lambda (A):quad |lambda -mu |leq |delta A|_{2}} Or in alternate formulation: {displaystyle exists lambda in Lambda (A):quad left|lambda -lambda ^{a}right|leq {frac {|{boldsymbol {r}}|_{2}}{left|{boldsymbol {v}}^{a}right|_{2}}}} which obviously remains true if A is a Hermitian matrix. In this case, however, 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(A) 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. Math. 2 (1): 137–141. doi:10.1007/BF01386217. S2CID 121278235. Eisenstat, S. C.; Ipsen, I. C. F. (1998). "Three absolute perturbation bounds for matrix eigenvalues imply relative bounds". SIAM Journal on Matrix Analysis and Applications. 20 (1): 149–158. CiteSeerX 10.1.1.45.3999. doi:10.1137/S0895479897323282. hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Spectral theoryTheorems in analysis