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Stability Analysis, Convex Hulls of Matrix Powers and Their STABILITY ANALYSIS, CONVEX HULLS OF MATRIX POWERS AND THEIR RELATIONS TO P-MATRICES By PATRICK SCOTT KISA TORRES A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics and Statistics MAY 2018 c Copyright by PATRICK SCOTT KISA TORRES, 2018 All Rights Reserved c Copyright by PATRICK SCOTT KISA TORRES, 2018 All Rights Reserved To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of PATRICK SCOTT KISA TORRES find it satisfactory and recommend that it be accepted. Michael Tsatsomeros, Ph.D., Chair Judith McDonald, Ph.D. Mark Schumaker, Ph.D. ii ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor, Professor Michael J. Tsatsomeros, for all the guidance, encouragement, patience, and advice that he has given me throughout my time as his student. I have been extremely fortunate to have an advisor who genuinely cared about my ideas and provided excellent feedback on my work. My collaboration with him has been a pleasant experience because of his love and enthusiasm for the field of Matrix Theory. I will always be grateful for all that he has taught me and for inspiring me to evolve as a mathematician. I would like to thank my committee members, Professor Judith McDonald and Professor Mark Schumaker, for all their support over the years and for raising a number of interesting points of discussion and possible future research directions during my dissertation defense. I would also like to thank Professor Lynn Schreyer and Professor Hong-Ming Yin for their constant support and valuable advice on teaching and research. I am eternally grateful for all the support that my parents, Herman and Maria, have given me in all of my pursuits. My accomplishments would not have been possible without their faith, love and encouragement. I would like to thank my sister Shylma, my brother T.J. and his wife Alicia for being supportive of me during all the years that I was a college student. I thank my niece Isa and my nephew Mason for always being a source of positive energy. I am extremely thankful to my best friend and fellow doctoral student, Ben Rapone, for his friendship and for helping me cope iii with life as a graduate student. I would like to thank Ben's wife Erin and their son Monroe for being great friends and for always being so supportive of me. My time in Pullman has been extra special because of the Rapone family. My life in Pullman has been met with a tremendous amount of support and encouragement from other members in the Department of Mathematics and Statistics. I particularly would like to thank Silvia Madrid, Diane Gilchrist, Kris Johnson, and Christy Jacobs for always taking the time to talk with me whenever I had a question or concern. Their support throughout my doctoral program and in all of my future endeavors means so much to me. It is a pleasure for me to extend my gratitude to my professors in the Department of Applied Mathematics at the University of Washington for their support during and after my studies there as a masters student. I particularly would like to thank Professor Mark Kot and Professor Nathan Kutz for their recommendations for me to pursue graduate work in pure and applied mathematics. I would also like to thank Professor Anne Greenbaum for being the first to introduce me to the field of Matrix Theory. My short time working with her in the summer of 2009 motivated me to pursue doctoral research in this area. Finally, I would like to thank my former math teachers and colleagues at South Seattle College for their many years of support, especially my longtime friend and fellow mathematician, Heidi Lyman. It was her love and passion for mathematics that inspired me to follow in her footsteps. iv STABILITY ANALYSIS, CONVEX HULLS OF MATRIX POWERS AND THEIR RELATIONS TO P-MATRICES Abstract by Patrick Scott Kisa Torres, Ph.D. Washington State University May 2018 Chair: Michael Tsatsomeros Invertibility of all convex combinations of an n × n matrix A and the n × n identity matrix I is equivalent to the real eigenvalues of A, if any, being positive. Moreover, 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). In this dissertation, we extend these results 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 open right half-plane and provides a general context for the theory of matrices with P-matrix powers. We present a new result regarding the spectrum of A that holds as a consequence of assuming the invertibility of all infinite convex combinations of A and its powers. v TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................................. iii ABSTRACT...............................................................v SYMBOLS................................................................. viii 1. INTRODUCTION.....................................................1 1.1 Motivation.........................................................1 1.2 Outline of the Dissertation.........................................1 2. THE CONCEPT OF STABILITY......................................4 2.1 Introduction.......................................................4 2.2 Stability in Autonomous Continuous Time Systems.................5 2.3 Positive Stable Matrices............................................8 3. P-MATRICES......................................................... 16 3.1 Introduction....................................................... 16 3.2 Basic Properties of P-matrices...................................... 17 3.3 Applications of P-matrices......................................... 26 3.4 Relations to Positive Stability...................................... 28 3.5 Open Problems.................................................... 33 3.6 Relations to Convex Combinations of Rows of Matrices............. 35 4. GENERALIZED STABILITY.......................................... 42 4.1 Extension to Arbitrary Powers of Matrices.......................... 43 4.2 The Power Series Case............................................. 46 5. EXTENSION TO INTERVAL MATRICES............................. 50 5.1 Basic Definitions................................................... 50 vi 5.2 Necessary and Sufficient Regularity Conditions..................... 52 5.3 Hadamard Products of Matrix Powers.............................. 60 6. CONCLUSIONS....................................................... 62 BIBLIOGRAPHY.......................................................... 65 vii SYMBOLS R, Rn set of real numbers and vectors C set of complex numbers Z set of integers Mn(R) set of n × n real matrices Mn(C) set of n × n complex matrices GLn(R) set of n × n real invertible matrices GLn(C) set of n × n complex invertible matrices Pn(R) set of n × n real P-matrices Pn(C) set of n × n complex P-matrices R+ nf0g positive real axis AT , A∗ transpose and conjugate transpose of A I identity matrix J all-ones square matrix σ(·) spectrum of a square matrix ρ(·) spectral radius of a square matrix ◦ Hadamard (i.e. entrywise) product for matrices hni f1; 2; : : : ; ng where n is a positive integer jαj cardinality of α ⊆ hni A[α] principal submatrix of A indexed by α ⊆ hni A[αjβ] submatrix of A whose rows are indexed by α and columns by β viii Arg z principal value (argument) of z 2 C 2 P2 2−j P2 Ct A 2 Mn(R) j=0 tjA 2 GLn(R) 8tj 2 [0; 1]; j=0 tj = 1 2 P2 2−j P2 CT A 2 Mn(R) j=0 TjA 2 GLn(R) 8Tj 2 [0;I]; j=0 Tj = I 2 2 Pt A 2 Mn(R) t0A + t1A 2 Pn(R) 8t0; t1 2 [0; 1]; t0 + t1 = 1 2 2 P A 2 Mn(R) A; A 2 Pn(R)g nPk k−j Pk o Pk(z) j=0 tjz tj 2 [0; 1]; j=0 tj = 1 nPk k−j Pk o Fk(z) j=0 Tjz Tj 2 [0;I]; j=0 Tj = I nPk k−j Pk o Gk(z) j=0 Sjz Sj 2 [0;J]; j=0 Sj = J Ct(n; k; R) A 2 Mn(R) f(A) 2 GLn(R) 8f 2 Pk(z) CT(n; k; R) A 2 Mn(R) F (A) 2 GLn(R) 8F 2 Fk(z) n Pk k−jo CS(n; k; R) A 2 Mn(R) G(A) 2 GLn(R) 8G 2 Gk(z);G(A) = j=0 Sj ◦ A n P1 j P1 o P1 p(z) = j=0 tjz tj 2 [0; 1]; j=0 tj = 1 0 P1 p 2 P1 a finite number of the tj's are nonzero π π Wk z 2 C n f0g − k < Arg z < k ; k 2 Z; k ≥ 2 Int(A; B) interval matrix of square matrices A and B AI Int(A; A) where A ≤ A ≤ A entrywise ix Dedication This dissertation is dedicated to my beloved parents, Herman and Maria, for their unconditional love, endless support, and unwavering encouragement. x \The Matrix is everywhere. It is all around us." − Morpheus (Laurence Fishburne) xi CHAPTER 1. INTRODUCTION 1.1 Motivation The location of the eigenvalues of an n×n real matrix A 2 Mn(R), the invertibility of the matrices in certain convex hulls constructed from A, and the positivity of the principal minors of A are three notions that are intimately related. Such relations are examined, e.g., in [23, 24, 29, 30, 31, 33, 38]. A key role is played by P-matrices, that is, matrices all of whose principal minors are positive, and a typical concern in many of these analyses is whether or not all the eigenvalues of a matrix lie in the open right half-plane. This dissertation is motivated by the following conjecture: If A and A2 are P-matrices, then all of the eigenvalues of A lie in the right half- plane. Our results lead to a necessary condition for which this conjecture may be true. 1.2 Outline of the Dissertation The outline of this dissertation is as follows.
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