Extension of Brauer and Rado perturbation theorems for regular matrix pencils

Abstract

In this paper, we propose new results for changing eigenvalues of a regular matrix pencil A − λ B, which are based on the well-known Brauer’s theorem [A Brauer, Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 19, 75-91, 1952] and Rado’s theorem [B N Parlett, Symmetric matrix pencils, J. Comput. Appl. Math., 38, 373-385, 1991.]. These results allow us to change eigenvalues of the original matrix pencil without altering its regularity and in a quite simple way, even allowing to change infinite eigenvalues. We also present an extension of Rado’s theorem that allows changing eigenvalues of a regular symmetric matrix pencil without altering its symmetric structure, and we show how to use these results in order to change the eigenvalues of a quadratic polynomial matrix. Finally, we present numerical examples that confirm the expected results with the new extensions of these theorems.