Department of Mathematics and Statistics
Washington State University
Title: Stability and Convex Hulls of Matrix Powers
Abstract: Invertibility of all
convex combinations of a square matrix A and the identity matrix I is
equivalent to the real eigenvalues, if any, being positive.
Invertibility of all matrices whose rows are convex combinations of the
respective rows of A and I is equivalent to A having positive principal
minors (i.e., being a P-matrix). These results are extended by
considering convex combinations of higher powers of A and of their
rows. The invertibility of matrices in these convex hulls is associated
with the eigenvalues of A lying in open sectors of the right-half plane
and provides a general context for the theory of matrices with P-matrix
powers. We wish to further extend these results to interval matrices
and determine additional restrictions on their spectra.