Bauer–Fike 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.

In what follows we assume that:

Proof. We can suppose μ ∉ Λ(A), otherwise take λ = μ and the result is trivially true since κ_p(V) ≥ 1. Since μ is an eigenvalue of A + δA, we have det(A + δA − μI) = 0 and so

However our assumption, μ ∉ Λ(A), implies that: det(Λ − μI) ≠ 0 and therefore we can write:

This reveals −1 to be an eigenvalue of

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:

But (Λ − μI)⁻¹ is a diagonal matrix, the p-norm of which is easily computed:

whence:

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, (λ^a, v^a ) and needs to bound the error. The following version comes in help.

Proof. We can suppose λ^a ∉ Λ(A), otherwise take λ = λ^a and the result is trivially true since κ_p(V) ≥ 1. So (A − λ^aI)⁻¹ exists, so we can write:

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