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.

Informally speaking, what it says is that the sensitivity of the eigenvalues is estimated by the condition number of the matrix of eigenvectors.

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, (λa, va ) and needs to bound the error.

Applying the Bauer–Fike theorem to Aa + (δA)a with eigenvalue 1, gives us: If A is normal, V is a unitary matrix, therefore: so that κ2(V) = 1.