Bauer-Fike Theorem

The Bauer-Fike theorem, published by Friedrich L. Bauer and C. T. Fike in 1960, is a result in matrix perturbation theory giving an absolute upper bound on how far an eigenvalue of a perturbed diagonalizable matrix can deviate from the nearest eigenvalue of the original, unperturbed matrix. The bound is expressed through the condition number of the matrix of eigenvectors, so that a matrix whose eigenvectors are close to being linearly dependent has eigenvalues that are correspondingly more sensitive to perturbation. The theorem is a standard tool in numerical analysis for judging how reliable a computed eigenvalue is once rounding error or other small perturbations are taken into account.