A secular equation for the eigenvalues of a diagonal matrix perturbation

Let D denote a diagonal n x n complex matrix, and suppose x(1), ..., x, and w(1), ..., w(r) are complex n-vectors. It is shown that there is a rational function F such that if lambda is not an eigenvalue for D, then lambda is an eigenvalue for P = D + x(1)*w(1) + ... +x(r)*w(r) if and only if F(lambda) = 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, lambda is an eigenvalue of multiplicity m > 0 for P if and only if F(lambda) = F'(lambda) = ... = F-(m-1)(lambda) = 0 and F-(m)(lambda) not equal 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.