Computing the Eigenvalues and Eigenvectors of Arrowhead Matrices

This paper treats the eigenvalue problem for a symmetric matrix which is zero except for its main diagonal and one row and column. Such problems arise in the description of radiationless transitions in isolated molecules and of oscillators vibrationally coupled with a Fermi liquid. In these applications the order n of the matrix A can be in the thousands. The purpose of this paper is to present formulas and efficient algorithms for computing eigenvalues and eigenvectors of such matrices.