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 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:
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:
since A is diagonalizable; taking the p-norm of both sides, we obtain:
is a diagonal matrix and its p-norm is easily computed:
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:
Note.||A−1δA|| can be formally viewed as the relative variation ofA, 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:
If we set:
then we have:
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:
Bauer, F. L.; Fike, C. T. (1960). "Norms and Exclusion Theorems". Numer. Math. 2 (1): 137–141. doi:10.1007/BF01386217.
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. doi:10.1137/S0895479897323282.