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On Certain Sets of Matrices: Euclidean Squared Distance Matrices, Ray-Nonsingular Matrices and Matrices Generated by Reflections W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 2004 On certain sets of matrices: Euclidean squared distance matrices, ray-nonsingular matrices and matrices generated by reflections Thomas W. Milligan College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Mathematics Commons Recommended Citation Milligan, Thomas W., "On certain sets of matrices: Euclidean squared distance matrices, ray-nonsingular matrices and matrices generated by reflections" (2004). Dissertations, Theses, and Masters Projects. Paper 1539623440. https://dx.doi.org/doi:10.21220/s2-0qeq-ag71 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]. ON CEBTAJN SETS OF MATEICSS Euclidean squared distance matrices, ray-nonsingular matrices and matrices generated by reflections A Dissertation Presented to The Faculty of the Department of Applied Science The College of William and Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Thomas W. Milligan April 2004 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I'lis Dissri'uitioij if submitted in pjirtiai fiilfillnient of Uk ' requirements for the Degree of Doctor of Pliik)so}jli,y _iYy tliomas W. Milligan A.ppruved by the Committee, April 2004 : 7 . Chi-Kwong |ii 5„plutii o'v\iViathias freg Smith .... Michael csatsomeros D('|>arfcment of Mathematics Washington State University 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I would like to d«licate this dissertation to m j wife, Amy. W ithout your love and support, I would never reach my dreams. And to my children, David, Alexandra, Megan and Nathaniel. Sometimes it was only by watching you grow and learn that I remember that learning is fun and to ENJOY my "work”. m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Acknowledgements vi List of Figures vil List of Tables viii Abstract ix 1 Introduction 2 1.1 Notation and Definition .............................................................................. 2 1.2 Overview............................................................................................................ 4 2 Euclidean Squared Distance Matrices 7 2.1 Characterization of ESD Matrices .............................................................. 8 2.2 Consequences of the Characterizations of V { n ) ....................................... 27 2.3 Uniqueness of Completions ........................................................................... 42 2.4 Spherical ESD Matrices .............................................................................. 58 3 Ray-Nonsingular Matrices 74 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 Introduction ............................................. 74 3.2 Preliminary Results and Strategies of Proof . ..................................... 75 3.3 Sufficient Conditions for 3 x .3 Patterns to be Strongly Balanceable . 78 3.4 Proof of Main Theorem ................................................................................. 83 3.5 Figures arid T a b le s ........................................................................................ 115 4 Finite Reflection Groups 121 4.1 Introduction and Background .................................................................... 121 4.2 Preliminary R esu lts ........................................................................................ 124 4.3 H a ......................................................................................................................... 126 4.4 H 4 ......................................................................................................................... 131 4.5 F 4 ......................................................................................................................... 139 4.6 E g ..................................................................................................................... 145 4.7 E 7 ......................................................................................................................... 152 4.8 E g ......................................................................................................................... 160 4 . 9 MATLAB Programs .................................................................................... 168 Bibliography 178 V ita 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements 1 would like to thaiik my co-advisors, Roy Mathias and Chi-Kwong Li, for their guidance and encouragement. I have enjoyed the insight, assistance and constructive criticism of Chi-Kwoug Li these last couple years. I would also like to thank Wayne Barret for introducing me to the exciting and enjoyable areaof matrix theory. I would also like to thank Michael Ti’osset for introducing me to distance matrices and for his help, his guidance and his time during our collaboration. Thanks are also due to Bryan Shader for his assistance and direction during our collaboration. Not to be forgotten, I appreciate the time spent reviewing this dissertation by my committee, which included ray advisors Chi-Kwong Li and Roy Mathias as well as Greg Smith from the Applied Sciiice Department and Michael Tsatsomeros from Washington State University. I would also like to acknowledge my support under an NSF grant. VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 3.1 GirapMcal Representation of J?(0,0 ) ........................................................... 79 3.2 Graphical Representation of R{a, 0 ) ........................................................... 79 3.3 Representations of R(0,0) n R{a, P) for Forms (1) - ( 4 ) ..................... 115 3.4 Representations of i?,(0,0) n R{a,f^) for Forms (5) ( 1 0 ) .................... 116 3.5 Representations of R(0,0) n R(q !,;9) for Forms (11) - ( C 4 ) ................ 117 3.6 Representations of ,fi’(0,0) fl R (a ,0 ) for B‘'orms (Co) (C 1 2 ) .................. 118 Vll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 3.1 Classes of vectors ............................................................................................. 81 3.2 Intersection of solution sets ............................................... 119 3.3 Conditions for empty intersectioxr of the two solution sets ...........................120 vni Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract In this dissertation, we study three different sets of matrices, First, we consider Euclidean distance squared matrices. Given n points in Euclidean space, we construct an n X n Euclidean squared distance m atrix by assigning to each entry the square of the pairwise interpoint Euclidean distance. The study of distance matrices is useful in computational chemistry and structural molecular biology. The purpose of the first part of the thesis is to better understand this set of matrices and its different characterizations so that, a number of open problems migiit be aixswered and known results improved. We look at geometrical propertitxs of this set, investigate forms of linear maps that preserve this set, consider the uniqueness of completions to this set and look at subsets that form regular figures. In the second part of this thesis, we consider ray-pattem matrices. A ra,y-pattern matrix is a complex matrix with each nonzero entry having modulus one. A ray-pattern is said to be ray~nonsiiigular if all positive entry-wise scalings are nonsingular. A lull ray-pattern matrix has no zero entries. It is known that for n > 5, there are no full ray-nonsingular matrices but examples exist for n < 5. We show that there are no 5 x 5 full ray- nonsingular matrice.s. The last part of this thesis studies certain of the finite reflection groups. A reflection is a linear endomorphism T on the Euclidean space V such that T{v) = n — 2(u, u)u for all v G V. A reflection group is a group of invertible opei’ators in the algebra of linear endomorphism on V that are generated by a set of reflections. One question that has recently been studied is the form of linear operators that preserve finite reflection groups. We first discuss known results about preservers of some finite reflection groups. We end by showing the forms of the remaining open cases. IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ON CERTAIN SETS OF MATRICES Euclidean squared distance matrices, Ray-nonsingular matrices and matrices generated by reflections Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction The theme of this dissertation is the study of certain sets of matrices. We will consider different open problems for each of these sets. To begin, we will first give some basic definitions and notation. Then we will give an overview
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