CLaN Research Group | CLaN
Seminar | WSU Colloquium | Department Home

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

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.

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.