The eigenvalue problem for ‘‘arrow’’ matrices
Olaf Walter, Lorenz S. Cederbaum, J. Schirmer · Journal of Mathematical Physics · 1984
The eigenvalue problem for a particular class of ‘‘arrow’’ matrices Z=(ABB†γC), where A is a Hermitian N×N matrix, C a real diagonal M×M matrix, B an arbitrary complex N×M matrix, and γ a real number, is investigated by means of a partitioning technique. Both Hermitian (γ=1) and non-Hermitian (γ≠1) arrow matrices Z are studied. The one-dimensional case (dimension N of A equal to 1) is briefly reviewed and a detailed treatment of the multidimensional case (N>1) is presented. For Hermitian arrow matrices, the analysis leads to a new algorithm for computing the eigenvalues and eigenvectors of Z which is particularly efficient if M≫N.