A note on bounds to the variation of eigenvalues in symmetric matrix perturbation of rank one
Jacques Bénasséni · Linear and Multilinear Algebra · 1990
Consider the perturbation A + B of a square matrix A by a rank-one matrix B. In various situations, it is useful to assess the effects of this perturbation on the eigenvalues or singular values of A. Different aspects have already been tackled in references [1] to [5] and this paper aims to underline two further properties. More specifically, assume that A is a n × n real symmetric matrix with distinct eigenvalues and that its eigenvalues and eigenvectors have been calculated. Also assume that B is a n × n real symmetric matrix of rank one which can therefore be written as B = ± uu′ for some column vector u of and let C = A + B for notational convenience. Then, the eigenvalues of any n × n real symmetric matrix M being denoted in this paper by , a classical result derived from the Courant-Fisher theorem gives for i= 1, …n with the euclidean norm . We suggest two simple improvements, hoped to be new, to these classical inequalities.