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Washington State University
Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar


Department of Mathematics and Statistics
Neill Hall 5W
October 17, Monday, 4:10 - 5:00 PM


 

Patrick Torres

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.