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Totally Nonnegative Matrices W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 1999 Totally nonnegative matrices Shaun M. Fallat College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Mathematics Commons Recommended Citation Fallat, Shaun M., "Totally nonnegative matrices" (1999). Dissertations, Theses, and Masters Projects. Paper 1539623948. https://dx.doi.org/doi:10.21220/s2-khbp-m642 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. 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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproductionFurther reproduction prohibited without prohibited permission. without permission. NOTE TO USERS This reproduction is the best copy available UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproductionFurther reproduction prohibited without prohibited permission. without permission. Totally Nonnegative Matrices A Dissertation Presented to The Faculty of the Department of Applied Science The College of William &: Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Shaun M. Fallat April 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9942549 Copyright 1999 by Fallat, Shaun Michael All rights reserved. UMI Microform 9942549 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET This Dissertation is submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy / Shaun M.^atlat, Author’ APPROVED. April 1999 Charles R. Ji ron Roy C. M athias Ilya M. SpitkoysKy David P. Stanford Eugene R. Tpraey, Physics £ Michael I. Gekhtman £ Department of Mathematics University of Notre Dame n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation is lovingly dedicated to my wife, Darcie, for her unfailing support and encouragement throughout my academic career. You have stood proudly beside me every seep of the way, and faithfully supported all of my decisions ... Good or Bad! Put simply: You are the secret of my success ... iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o n ten ts Acknowledgments vi List of Figures vii Abstract viii 1 Introduction 2 1.1 Notation and Matrix Theoretic Background .............................................. 2 1.2 Certain Determinantal Identities..................................................................... 4 1.3 Overview................................................................................................................ 7 2 Preliminary Results and Discussion 14 2.1 Background R esu lts ............................................................................................. 15 2.2 Factorizations of Totally Nonnegative M atrices .......................................... 22 2.3 Classical Determinantal Inequalities ............................................................... 29 2.4 Altering Totally Nonnegative M atrices ........................................................ 31 2.5 Zero-Nonzero Patterns of Totally Nonnegative M atrices ......................... 36 2.6 Retractibility of Totally Nonnegative M atrices .......................................... 44 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7 Row and Column Inclusion .............................................................................. 51 2.S Sub-direct S u m s .................................................................................................. 5S 3 Spectral Properties and Jordan Structures 72 3.1 Main R e s u lt......................................................................................................... 76 3.2 Jordan Structures of Totally Nonnegative Matrices .............................. S3 4 Determinantal Inequalities 107 4.1 Preliminaries ..................................................................................................... 10S 4.2 Operators Preserving Bounded R a tio s.......................................................... 120 4.3 Special Ratio Types ........................................................................................... 129 4.4 General Case for n < 5 .................................................................................... 146 4.5 Summary and Further Discussion ................................................................ 159 5 Hadamard Products of Totally Nonnegative Matrices 164 5.1 Notation and General Set-up .......................................................................... 166 5.2 Elements of the Hadamard D ual ................................................................... 171 5.3 The 3-by-n Case .............................................................................................. 175 5.4 (Q,+)-Patterns..................................................................................................... 1S4 5.5 Oppenheim’s Inequality ................................................................................. 191 Bibliography 194 Vita 202 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I gratefully acknowledge the assistance of the entire Mathematics Department at the College of William and Mary for their willingness to treat and accept me as a colleague rather than a student. All of you have made this experience a pleasure and I thank you for that. In particular, I wish to thank my advisor, Charles Johnson, for introducing me to the wonderful world of totally nonnegative matrices and his enthusiasm in studying this topic. I would like to express my deepest thanks to my r > n™ J I?... ~ »-r r xi, r ..i ~ wxj-txju-i v uvg ixM JXixucxd. ilu j , xxy ct, j-scbvxvx tm u x-iu.gc.nc: xx iu i 0-u.cj.x v-clx<jxux xx-ctnxxig ut my thesis and subsequent recommendations. Finally, I am grateful to my external examiner, Michael Gekhtman, for his assistance (as some of this dissertation is joint work) and support throughout this entire process. VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 2.1 Elementary Diagram ............................................................................................ 25 2.2 Second Elementary Diagram .............................................................................. 26 2.3 Diagonal Elementary Diagram .......................................................................... 27 0 A. QpnAral 2_Ky _3 D1-qT 2IH. 2^ 2.5 General n-by-n diagram ...................................................................................... 29 3.1 Fixed Principal Rank One .................................................................................. S7 3.2 Fixed Principal Rank Two ................................................................................. S8 3.3 Principal Rank Two ............................................................................................. 98 4.1 General n-by-n diagram ...................................................................................... I l l 4.2 3-by-3 diagram ......................................................................................................
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