Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1999 The intersection of some classical equivalence classes of matrices Mark Alan Mills Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Mills, Mark Alan, "The intersection of some classical equivalence classes of matrices " (1999). Retrospective Theses and Dissertations. 12153. https://lib.dr.iastate.edu/rtd/12153 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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UMI 300 North Zeeb Road Ann Arbor, MI 48103 ii Graduate College Iowa State University This is to certify that the Doctoral dissertation of Mark Alan Mills has met the dissertation requirements of Iowa State University Signature was redacted for privacy. Signature was redacted for privacy. Signature was redacted for privacy. For th ^^ uate College iii TABLE OF CONTENTS ACKNOWLEDGEMENTS v ABSTRACT vi CHAPTER 1 INTRODUCTION AND BACKGROUND 1 1.1 Equivalence Classes 1 1.2 Canonical Forms and Invariants 2 1.3 Unitary Similarity 5 1.4 The Problem 6 CHAPTER 2 THE CONJUNCTIVE-SIMILARITY EQUIVALENCE CLASS 7 2.1 CS(.4) and U[A] 7 2.2 CS(.4): The n xra Case (Part 1) 8 2.3 CS(.4): The 1 X 1 Case 12 2.4 CS(.4): The 2 X 2 Case 12 2.5 CS(.4): The n x ra Case (Part 2) 16 2.5.1 Sim(.4), Conj(.4), and ZY(A) 16 2.5.2 Diagonal T and Triangular .4 17 2.5.3 Direct Sum of 2 x 2 and Identity Matrices 18 2.5.4 AM Similar to .4 18 2.5.5 .4 in Jordan Canonical Form 21 2.5.6 Conclusions 21 2.6 Invariants of CS(.4) 21 2.7 The Converse Problem 23 iv CHAPTER 3 THE SIMILARITY-UNITARY EQUIVALENCE EQUIVA­ LENCE CLASS 27 3.1 UES(.4) and U{A) 27 3.2 UES(.4): The n x n Case 28 3.3 UES(.4): The 1 x 1 Case 34 3.4 UES(.4): The 2x2 Case 35 3.5 UEquiv(A) and U{A) 35 CHAPTER 4 CONCLUSIONS AND FUTURE DIRECTIONS 37 4.1 CS(A) 37 4.2 UES(4) 37 APPENDIX 39 BIBLIOGRAPHY 41 V ACKNOWLEDGEMENTS First, I thank God for His guidance throughout this process and for the abilities and strength He has given me to be able to reach this far. When I consider God's creation of mathematics, Psalm 19:1-4 (New International Version) is certainly correct when it says: "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they display knowledge. There is no speech or language where their voice is not heard. Their voice goes out into all the earth, their words to the ends of the world." Next, I thank the many family members and friends who have supported and encouraged me during this time. Their confidence in me allowed me to reach further than I ever thought I could. I am especially indebted to my parents for their support (both emotional and monetary) that has allowed me to achieve a dream. Without them, none of this would have been possible. Finally, I thank Dr. Bryan Cain for his mathematical guidance which led to this disser­ tation. His initial interest in this problem and his insights, encouragement, and friendship throughout this process have been invaluable to me. vi ABSTRACT Let A be an n X ra complex matrix. Let Sim (A) denote the similarity equivalence class of A, Conj(.4) denote the conjunctivity equivalence class of .4, UEquiv(.4) denote the unitary- equivalence equivalence class of .4, and 2/{{A) denote the unitary similarity equivalence class of A. Each of these equivalence classes has been studied for some time and is generally well- understood. In particular, canonical forms have been given for each equivalence class. Since the intersection of any two equivalence classes of .4 is again an equivalence class of .4, we consider two such intersections: CS(.4) = Sim(.4) fl Conj(.4) and UES(.4) = Sim(A) n UEquiv(.4). Though it is natural to first think that each of these is simply U{A), for each .4. we show by examples that this is not the case. We then try to classify which matrices .4 have CS(.4) = U{A). For matrices having CS(.4) ^ 1({A), we try to count the number of disjoint unitary similarity classes contained in CS(.4). Though the problem is not completely solved for CS(.4). we reduce the problem to non-singular, non-co-Hermitian matrices .4. A similar analysis is performed for UES(.4), and a (less simple) reduction of the problem is also achieved. 1 CHAPTER 1 INTRODUCTION AND BACKGROUND 1.1 Equivalence Classes Throughout mathematics, the notion of an equivalence class is a basis for understanding the structure of various algebraic objects (e.g., cosets in group theory). If 5 is a set, then recall that /? C 5 X 5 is an equivalence relation on 5 provided R is: reflexive : (a, a) € i? for all a 6 S: symmetric : (a, b) e B. => {b. a) € R; and transitive : (a, 6) € R and (6, c) € i? (a, c) e R- Two elements a, 6 6 S are said to be equivalent, if (a, b) 6 R- We can then define the equivalence class of a particular element a £ S as the set of all elements in 5 that are equivalent to a via the equivalence relation R. If there are two different equivalence relations Ri and R2 on S, then, for a fixed element a 6 5, the intersection of the two equivalence classes of a is again an equivalence class of a with respect to the new equivalence relation n i?2 on 5. The set S is the union of the equivalence classes of a (possibly small) subset of the elements in 5, and this gives S some structure. Certainly this is true in group theory, where a group is the union of its cosets, which are equivalence classes of a particular subset of group elements. This is also true for the set of all n x n complex matrices (denoted by Mn), where there are a number of well-known equivalence relations. Recall that two matrices A,B£ M„ are said to be similar if there exists an n x n invertible matrix 5 so that B = S~^AS. (I will denote the set of n x n complex invertible matrices by GLn.) Similarity is an equivalence relation in Mn, and we can consider the similarity 2 equivalence class of a matrix A, denoted by Sim(.4). If A and B are similar, then they have the same eigenvalues (counting algebraic multiplicities). (From here on, the term multiplicity when referring to eigenvalues will refer to algebraic multiplicity, not geometric multiplicity.) foil fool However, the converse is not true, as is shown by the matrices and that L0 0J [00. both have the eigenvalue 0 with algebraic multiplicity 2, but are not similar. Two matrices .4, B € Mn are said to be conjunctive (or 'congruent or Hermitian congruent), if there exists T € GLn so that B = T'AT, where T' denotes the conjugate transpose of the matrix T. Conjunctivity is an equivalence relation in Mn, and we can consider the conjunctive equivalence class of a matrix A, denoted by Conj(,4).