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Washington State University

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics

Neill Hall 5W

November 16, Monday, 4:10 - 5:00 PM

Patrick Torres

Department of Mathematics

Washington State University

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics

Neill Hall 5W

November 16, Monday, 4:10 - 5:00 PM

Patrick Torres

Department of Mathematics

Washington State University

Title: Convex Hulls of Matrices and Stability

Abstract: Nonsingularity of all convex combinations of a real square matrix A and the identity matrix I is equivalent to the spectrum of A containing no negative (real) eigenvalues. Moreover, nonsingularity of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A being a P-matrix (i.e. a matrix whose principal minors are all positive). We wish to extend these results by considering convex combinations (of the rows) of A^2, A, and I. The nonsingularity of these convex hulls is associated with the eigenvalues of A lying in the open right half of the complex plane (positive stability). This relationship provides a general context for many results and conjectures about the positive stability of matrices with P-matrix powers. In this talk, I will present some recent progress made in this research.

This talk is based on joint work with my advisor, Dr. Michael J. Tsatsomeros.

Abstract: Nonsingularity of all convex combinations of a real square matrix A and the identity matrix I is equivalent to the spectrum of A containing no negative (real) eigenvalues. Moreover, nonsingularity of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A being a P-matrix (i.e. a matrix whose principal minors are all positive). We wish to extend these results by considering convex combinations (of the rows) of A^2, A, and I. The nonsingularity of these convex hulls is associated with the eigenvalues of A lying in the open right half of the complex plane (positive stability). This relationship provides a general context for many results and conjectures about the positive stability of matrices with P-matrix powers. In this talk, I will present some recent progress made in this research.

This talk is based on joint work with my advisor, Dr. Michael J. Tsatsomeros.