A secular equation for the eigenvalues of a diagonal matrix perturbation

Abstract

Let D denote a diagonal n × n complex matrix, and suppose x 1, …, x r and w 1, …, w r are complex n-vectors. It is shown that there is a rational function F such that if λ is not an eigenvalue for D, then λ is an eigenvalue for P = D + x∗ 1w 1 + … + x∗ rw r if and only if F( λ) = 0. This generalizes a well-known result for the eigenvalues of a rank one self-adjoint perturbation. An immediate corollary in the rank one self-adjoint case is that the eigenvalues of P and D must interlace if the eigenvalues of D are distinct and the perturbation matrix is irreducible. It is shown that in the general case the function F also carries information about the eigenvalues of P. For example, λ is an eigenvalue of multiplicity m > 0 for P if and only if F( λ) = F′( λ) = … = F ( m − 1) ( λ) = 0 and F ( m) ( λ) ≠ 0. In the self-adjoint case, a necessary and sufficient condition for the eigenvalues of P and D to interlace is given, and the problem of determining the multiplicities of the eigenvalues of D as eigenvalues of P is studied. The formula yields a simple algorithm for determining the characteristic polynomial of a tridiagonal matrix.