A secular equation for the eigenvalues of a diagonal matrix perturbation
For a diagonal matrix \(D\) with complex entries \(\alpha_i\), the matrices \(P=D + x_1^*w_1+ \cdots +x^*_rw_r\), and the existence of a secular equation \(F(\lambda)\) are considered. An algorithm is presented to construct \(F\). It is shown that if \(\lambda\) is not an eigenvalue for \(D\), then \(\lambda\) is an eigenvalue for \(P\) if and only if \(F(\lambda) =0\). In the rank one self-adjoint perturbation the eigenvalues of \(P\) and \(D\) must interlace if the eigenvalues of \(D\) are distinct and the perturbation matrix is irreducible. The eigenvalues may be complex and of any multiplicity. An algorithm for determining the secular formula for a tridiagonal matrix is presented. The case of the self-adjoint perturbation of \(D\) is studied. The determination of the multiplicities of the eigenvalues of \(D\) as eigenvalues of \(P\) is discussed.
scientific article; zbMATH DE number 951790