DOCSLIB.ORG

JULIUS, HAYDEN, August 2021 PURE MATHEMATICS NONSTANDARD SOLUTIONS of LINEAR PRESERVER PROBLEMS (111 Pages) Dissertation Advisor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
JULIUS, HAYDEN, August 2021 PURE MATHEMATICS NONSTANDARD SOLUTIONS of LINEAR PRESERVER PROBLEMS (111 Pages) Dissertation Advisor JULIUS, HAYDEN, August 2021 PURE MATHEMATICS NONSTANDARD SOLUTIONS OF LINEAR PRESERVER PROBLEMS (111 pages) Dissertation Advisor: Mikhail Chebotar Linear preserver problems concern the characterization of linear operators on matrix spaces that leave invariant certain functions, subsets, relations, etc. We present several linear preserver prob- lems whose solutions may be considered nonstandard since they differ significantly from classical results. In addition, we also discuss several related linear preserver problems with standard solu- tions to highlight the phenomena observed. NONSTANDARD SOLUTIONS OF LINEAR PRESERVER PROBLEMS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Hayden Julius August 2021 c Copyright All rights reserved Except for previously published materials Dissertation written by Hayden Julius B.S., Cleveland State University, 2016 M.S., Kent State University, 2019 Ph.D. Kent State University, 2021 Approved by Mikhail Chebotar , Chair, Doctoral Dissertation Committee Joanne Caniglia , Members, Doctoral Dissertation Committee Feodor Dragan Mark L. Lewis Dmitry Ryabogin Accepted by Andrew M. Tonge , Chair, Department of Mathematical Sciences Mandy Munro-Stasiuk , Interim Dean, College of Arts and Sciences TABLE OF CONTENTS TABLE OF CONTENTS.................................. iv PREFACE........................................... vi ACKNOWLEDGMENTS.................................. ix 1 An introduction to linear preserver problems.................... 1 1.1 Types of linear preserver problems............................1 1.1.1 Subset preservers..................................1 1.1.2 Function preservers.................................5 1.1.3 Relation preservers.................................7 1.1.4 Transformation preservers............................. 10 1.2 Generalizations....................................... 12 1.3 Nonstandard solutions of linear preserver problems................... 13 2 Maps preserving certain elements of a subset.................... 17 2.1 On maps sending rank-k square-zeros to square-zeros.................. 20 2.2 On maps sending rank-t idempotents to idempotents.................. 24 2.3 Related problems and conjectures............................. 31 3 Maps preserving nonzero Lie products........................ 37 3.1 Commutativity preservers and Herstein's Lie isomorphism problem.......... 42 3.2 On maps preserving Lie products equal to e12 ...................... 45 3.2.1 Some results on triangularization......................... 45 iv 3.2.2 The main result................................... 47 3.3 On maps preserving Lie products equal to e11 − e22 ................... 57 3.4 Related problems and conjectures............................. 61 4 Maps preserving nonzero ordinary products..................... 64 4.1 Maps preserving products equal to fixed invertible matrices.............. 65 4.2 Maps preserving products equal to a diagonalizable matrix............... 69 4.3 Related problems and conjectures............................. 77 5 Maps preserving roots of nonhomogeneous polynomials.............. 84 5.1 Maps preserving the square roots of an idempotent matrix............... 89 5.2 Maps preserving the square roots of a rank-one nilpotent................ 92 BIBLIOGRAPHY ...................................... 98 v PREFACE The main goal of this dissertation is to illustrate the general framework of linear preserver problems (LPPs) and introduce new results that complement, generalize, or stand in contrast to existing theorems in the literature. However, LPPs form an extremely active research topic that incorporates objects and techniques from matrix theory, operator theory, noncommutative algebra, projective and algebraic geometry, functional analysis, etc., and in what follows, we will have need to discuss many preserver problems that touch on many of these areas. Consequently, I have decided in many places to state a known result in a simplified form and not in its full, optimal form. For instance, there are plenty of results below that hold for matrices over an arbitrary field or even on rectangular spaces of matrices, but it will be convenient to simply think of complex square matrices. Hence the theorems being represented here (more than likely) appear differently in their actual papers. There are close to 70 individual preserver problems mentioned in what follows, and thus it would be a significant task for the reader to keep track of each problem's minor differences. I will explicitly mention the full, optimal form if relevant to the matter at hand. I have also used uniform notation throughout. Hence the reader can usually expect the LPPs mentioned below to be stated on complex square matrices. The Greek letter φ is always reserved for the mapping between matrix spaces being discussed (unless a specific example has been defined). The hypotheses on φ might change from paragraph to paragraph and theorem to theorem. I have tried to make this as clear as possible. A list of fixed notations that I use throughout the document without reference is provided on page viii. Now I will also list the conventions used. Each vector in a finite-dimensional vector space over a field is represented as a column vector, and matrix multiplication is on the left. Furthermore, vi if T~ and S~ are two linear transformations with T and S their matrices relative to some ordered basis, then the matrix multiplication TS represents the composition T~S~, so that S~ \acts first” on the vector space. The full algebra of n × n matrices over a field F is denoted Mn(F ). I stipulate that whenever Mn(F ) is written, it is inherently assumed that n ≥ 2. All rings considered here are associative but do not necessarily have a multiplicative identity. A map that fixes the multiplicative identity is called a unital map. I have assumed a level of familiarity with linear algebra up to Jordan canonical forms and a basic understanding of groups, rings, and fields. vii Frequently used notation: C field of complex numbers Fp finite field of order p char(R) the characteristic of a ring R Mn(R) set of all n × n matrices over a ring R (shortened to Mn if R = C) sln(F ) set of trace-zero matrices over a field F (shortened to sln if F = C) dimF V dimension of a vector space V over a field F (shortened to dim V if F = C) (aij) a matrix with entries aij, where i indexes rows and j indexes columns eij matrix with 1 in the (i; j)-entry and zeros elsewhere (called a matrix unit) In n × n identity matrix, also written as e11 + e22 + ··· + enn At transpose of a matrix A tr(A) trace of a matrix A det(A) determinant of a matrix A rank(A) rank of a matrix A σ(A) spectrum of a matrix A A ◦ B Jordan product AB + BA [A; B] Lie product AB − BA C(A) centralizer of A, fB 2 Mn(F ) j AB = BAg FA set of F multiples of A, fcA j c 2 F g V1 ⊕ V2 direct sum of vector spaces φjV restriction of a map φ to V bxc the largest integer n such that n ≤ x dxe the smallest integer n such that x ≤ n viii ACKNOWLEDGMENTS Firstly I would like to express profound gratitude to my advisor Mikhail Chebotar. Misha's guidance over the past few years has been incredible. He has helped me publish a nontrivial amount of papers, navigate the academic job market during a global pandemic, and advocated for me when travel was not possible. In addition to his professional support, Misha has been there for me personally. I will fondly remember all of our conversations past and present about current events, fresh fish and coffee, the nature of academic publishing, and entertainment; in particular, stop motion cartoons about the Cold War. Thank you to each member of my defense committee, Mark Lewis, Dmitry Ryabogin, Feodor Dragan, and Joanne Caniglia for their generous time spent with my monstrous dissertation. I would also like to acknowledge Misha, Mark, and Jenya Soprunova for the wonderful opportunities with the 2019 and 2020 REU programs at Kent State University. Thank you to Victor Ginsburg, Ricardo Velasquez, Nikita Borisov, and Martha Sikora for their collaboration on problems that appear in this dissertation. I am honored to have played a (small) part in their future success as young researchers. I am especially grateful to Artem Zvavitch and Andrew Tonge for their guidance and support throughout my time at Kent State. Thank you to Patty Housley and Virginia Wright for their constant and invaluable assistance. I would also like to acknowledge my amazing fellow graduate students, past and present, who have been there for me over the years. Thank you to my parents, Mark and Alison, and my siblings, Maggie and Anthony, for their love and so much more. To Miranda, for everything. And to Toby, my best friend. ix CHAPTER 1 An introduction to linear preserver problems Linear preserver problems concern the characterization of linear operators on matrix spaces that leave invariant certain functions, subsets, relations, etc. Emerging prominently in the latter half of the 20th century, the earliest linear preserver problems arose naturally as converse problems from matrix theory. Since then, LPPs have become an active, fertile area of research with applications to many branches of mathematics, both pure and applied. What follows is a brief discussion highlighting several significant and classical results that moti- vate the upcoming presentation of results in later chapters. This introduction follows the expository papers [37, 52, 56]. 1.1 Types of linear preserver problems Let M denote a matrix space, by which we mean a vector space whose elements are linear operators. 1.1.1 Subset preservers Let S be a proper subset of M. We say that the linear map φ : M!M preserves S if A 2 S =) φ(A) 2 S and φ : M!M preserves S in both directions if A 2 S () φ(A) 2 S: A typical subset preserver problem asks for a characterization of maps φ such that S is invariant under φ. Equivalently, we write φ(S) ⊆ S if φ preserves S and φ(S) = S if φ preserves S in both 1 directions. In the familar case of Mn := Mn(C), there are many important subsets of matrices that are well understood and of chief interest in applications.
Load more

Recommended publications
  • Quantum Cluster Algebras and Quantum Nilpotent Algebras
    Quantum cluster algebras and quantum nilpotent algebras K. R. Goodearl ∗ and M. T. Yakimov † ∗Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A., and †Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. Proceedings of the National Academy of Sciences of the United States of America 111 (2014) 9696–9703 A major direction in the theory of cluster algebras is to construct construction does not rely on any initial combinatorics of the (quantum) cluster algebra structures on the (quantized) coordinate algebras. On the contrary, the construction itself produces rings of various families of varieties arising in Lie theory. We prove intricate combinatorial data for prime elements in chains of that all algebras in a very large axiomatically defined class of noncom- subalgebras. When this is applied to special cases, we re- mutative algebras possess canonical quantum cluster algebra struc- cover the Weyl group combinatorics which played a key role tures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results in categorification earlier [7, 9, 10]. Because of this, we expect for a large class of Poisson nilpotent algebras. Many important fam- that our construction will be helpful in building a unified cat- ilies of coordinate rings are subsumed in the class we are covering, egorification of quantum nilpotent algebras. Finally, we also which leads to a broad range of application of the general results prove similar results for (commutative) cluster algebras using to the above mentioned types of problems. As a consequence, we Poisson prime elements.
    [Show full text]
  • Introduction to Cluster Algebras. Chapters
    Introduction to Cluster Algebras Chapters 1–3 (preliminary version) Sergey Fomin Lauren Williams Andrei Zelevinsky arXiv:1608.05735v4 [math.CO] 30 Aug 2021 Preface This is a preliminary draft of Chapters 1–3 of our forthcoming textbook Introduction to cluster algebras, joint with Andrei Zelevinsky (1953–2013). Other chapters have been posted as arXiv:1707.07190 (Chapters 4–5), • arXiv:2008.09189 (Chapter 6), and • arXiv:2106.02160 (Chapter 7). • We expect to post additional chapters in the not so distant future. This book grew from the ten lectures given by Andrei at the NSF CBMS conference on Cluster Algebras and Applications at North Carolina State University in June 2006. The material of his lectures is much expanded but we still follow the original plan aimed at giving an accessible introduction to the subject for a general mathematical audience. Since its inception in [23], the theory of cluster algebras has been actively developed in many directions. We do not attempt to give a comprehensive treatment of the many connections and applications of this young theory. Our choice of topics reflects our personal taste; much of the book is based on the work done by Andrei and ourselves. Comments and suggestions are welcome. Sergey Fomin Lauren Williams Partially supported by the NSF grants DMS-1049513, DMS-1361789, and DMS-1664722. 2020 Mathematics Subject Classification. Primary 13F60. © 2016–2021 by Sergey Fomin, Lauren Williams, and Andrei Zelevinsky Contents Chapter 1. Total positivity 1 §1.1. Totally positive matrices 1 §1.2. The Grassmannian of 2-planes in m-space 4 §1.3.
    [Show full text]
  • Chapter 2 Solving Linear Equations
    Chapter 2 Solving Linear Equations Po-Ning Chen, Professor Department of Computer and Electrical Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C. 2.1 Vectors and linear equations 2-1 • What is this course Linear Algebra about? Algebra (代數) The part of mathematics in which letters and other general sym- bols are used to represent numbers and quantities in formulas and equations. Linear Algebra (線性代數) To combine these algebraic symbols (e.g., vectors) in a linear fashion. So, we will not combine these algebraic symbols in a nonlinear fashion in this course! x x1x2 =1 x 1 Example of nonlinear equations for = x : 2 x1/x2 =1 x x x x 1 1 + 2 =4 Example of linear equations for = x : 2 x1 − x2 =1 2.1 Vectors and linear equations 2-2 The linear equations can always be represented as matrix operation, i.e., Ax = b. Hence, the central problem of linear algorithm is to solve a system of linear equations. x − 2y =1 1 −2 x 1 Example of linear equations. ⇔ = 2 3x +2y =11 32 y 11 • A linear equation problem can also be viewed as a linear combination prob- lem for column vectors (as referred by the textbook as the column picture view). In contrast, the original linear equations (the red-color one above) is referred by the textbook as the row picture view. 1 −2 1 Column picture view: x + y = 3 2 11 1 −2 We want to find the scalar coefficients of column vectors and to form 3 2 1 another vector .
    [Show full text]
  • A Constructive Arbitrary-Degree Kronecker Product Decomposition of Tensors
    A CONSTRUCTIVE ARBITRARY-DEGREE KRONECKER PRODUCT DECOMPOSITION OF TENSORS KIM BATSELIER AND NGAI WONG∗ Abstract. We propose the tensor Kronecker product singular value decomposition (TKPSVD) that decomposes a real k-way tensor A into a linear combination of tensor Kronecker products PR (d) (1) with an arbitrary number of d factors A = j=1 σj Aj ⊗ · · · ⊗ Aj . We generalize the matrix (i) Kronecker product to tensors such that each factor Aj in the TKPSVD is a k-way tensor. The algorithm relies on reshaping and permuting the original tensor into a d-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many (1) (d) different structured tensors, the Kronecker product factors Aj ;:::; Aj are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors. Key words. Kronecker product, structured tensors, tensor decomposition, TTr1SVD, general- ized symmetric tensors, Toeplitz tensor, Hankel tensor AMS subject classifications. 15A69, 15B05, 15A18, 15A23, 15B57 1. Introduction. Consider the singular value decomposition (SVD) of the fol- lowing 16 × 9 matrix A~ 0 1:108 −0:267 −1:192 −0:267 −1:192 −1:281 −1:192 −1:281 1:102 1 B 0:417 −1:487 −0:004 −1:487 −0:004 −1:418 −0:004 −1:418 −0:228C B C B−0:127 1:100 −1:461 1:100 −1:461 0:729 −1:461 0:729 0:940 C B C B−0:748 −0:243 0:387 −0:243 0:387 −1:241 0:387 −1:241 −1:853C B C B 0:417
    [Show full text]
  • Introduction to Cluster Algebras Sergey Fomin
    Joint Introductory Workshop MSRI, Berkeley, August 2012 Introduction to cluster algebras Sergey Fomin (University of Michigan) Main references Cluster algebras I–IV: J. Amer. Math. Soc. 15 (2002), with A. Zelevinsky; Invent. Math. 154 (2003), with A. Zelevinsky; Duke Math. J. 126 (2005), with A. Berenstein & A. Zelevinsky; Compos. Math. 143 (2007), with A. Zelevinsky. Y -systems and generalized associahedra, Ann. of Math. 158 (2003), with A. Zelevinsky. Total positivity and cluster algebras, Proc. ICM, vol. 2, Hyderabad, 2010. 2 Cluster Algebras Portal hhttp://www.math.lsa.umich.edu/˜fomin/cluster.htmli Links to: • >400 papers on the arXiv; • a separate listing for lecture notes and surveys; • conferences, seminars, courses, thematic programs, etc. 3 Plan 1. Basic notions 2. Basic structural results 3. Periodicity and Grassmannians 4. Cluster algebras in full generality Tutorial session (G. Musiker) 4 FAQ • Who is your target audience? • Can I avoid the calculations? • Why don’t you just use a blackboard and chalk? • Where can I get the slides for your lectures? • Why do I keep seeing different definitions for the same terms? 5 PART 1: BASIC NOTIONS Motivations and applications Cluster algebras: a class of commutative rings equipped with a particular kind of combinatorial structure. Motivation: algebraic/combinatorial study of total positivity and dual canonical bases in semisimple algebraic groups (G. Lusztig). Some contexts where cluster-algebraic structures arise: • Lie theory and quantum groups; • quiver representations; • Poisson geometry and Teichm¨uller theory; • discrete integrable systems. 6 Total positivity A real matrix is totally positive (resp., totally nonnegative) if all its minors are positive (resp., nonnegative).
    [Show full text]
  • ON the PROPERTIES of the EXCHANGE GRAPH of a CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein 1. Main Defi
    Math. Res. Lett. 15 (2008), no. 2, 321–330 c International Press 2008 ON THE PROPERTIES OF THE EXCHANGE GRAPH OF A CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein Abstract. We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of A. Both conjectures were formulated recently by Fomin and Zelevinsky. 1. Main definitions and results A cluster algebra is an axiomatically defined commutative ring equipped with a distinguished set of generators (cluster variables). These generators are subdivided into overlapping subsets (clusters) of the same cardinality that are connected via sequences of birational transformations of a particular kind, called cluster transfor- mations. Transfomations of this kind can be observed in many areas of mathematics (Pl¨ucker relations, Somos sequences and Hirota equations, to name just a few exam- ples). Cluster algebras were initially introduced in [FZ1] to study total positivity and (dual) canonical bases in semisimple algebraic groups. The rapid development of the cluster algebra theory revealed relations between cluster algebras and Grassmannians, quiver representations, generalized associahedra, Teichm¨uller theory, Poisson geome- try and many other branches of mathematics, see [Ze] and references therein. In the present paper we prove some conjectures on the general structure of cluster algebras formulated by Fomin and Zelevinsky in [FZ3]. To state our results, we recall the definition of a cluster algebra; for details see [FZ1, FZ4].
    [Show full text]
  • Spectral Clustering and Multidimensional Scaling: a Unified View
    Spectral clustering and multidimensional scaling: a unified view Fran¸coisBavaud Section d’Informatique et de M´ethodes Math´ematiques Facult´edes Lettres, Universit´ede Lausanne CH-1015 Lausanne, Switzerland (To appear in the proccedings of the IFCS 2006 Conference: “Data Science and Classification”, Ljubljana, Slovenia, July 25 - 29, 2006) Abstract. Spectral clustering is a procedure aimed at partitionning a weighted graph into minimally interacting components. The resulting eigen-structure is de- termined by a reversible Markov chain, or equivalently by a symmetric transition matrix F . On the other hand, multidimensional scaling procedures (and factorial correspondence analysis in particular) consist in the spectral decomposition of a kernel matrix K. This paper shows how F and K can be related to each other through a linear or even non-linear transformation leaving the eigen-vectors invari- ant. As illustrated by examples, this circumstance permits to define a transition matrix from a similarity matrix between n objects, to define Euclidean distances between the vertices of a weighted graph, and to elucidate the “flow-induced” nature of spatial auto-covariances. 1 Introduction and main results Scalar products between features define similarities between objects, and re- versible Markov chains define weighted graphs describing a stationary flow. It is natural to expect flows and similarities to be related: somehow, the exchange of flows between objects should enhance their similarity, and tran- sitions should preferentially occur between similar states. This paper formalizes the above intuition by demonstrating in a general framework that the symmetric matrices K and F possess an identical eigen- structure, where K (kernel, equation (2)) is a measure of similarity, and F (symmetrized transition.
    [Show full text]
  • Orbital-Dependent Exchange-Correlation Functionals in Density-Functional Theory Realized by the FLAPW Method
    Orbital-dependent exchange-correlation functionals in density-functional theory realized by the FLAPW method Von der Fakultät für Mathematik, Informatik und Naturwissenschaen der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaen genehmigte Dissertation vorgelegt von Dipl.-Phys. Markus Betzinger aus Fröndenberg Berichter: Prof. Dr. rer. nat. Stefan Blügel Prof. Dr. rer. nat. Andreas Görling Prof. Dr. rer. nat. Carsten Honerkamp Tag der mündlichen Prüfung: Ô¥.Ôò.òýÔÔ Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. is document was typeset with LATEX. Figures were created using Gnuplot, InkScape, and VESTA. Das Unverständlichste am Universum ist im Grunde, dass wir es verstehen können. Albert Einstein Abstract In this thesis, we extended the applicability of the full-potential linearized augmented-plane- wave (FLAPW) method, one of the most precise, versatile and generally applicable elec- tronic structure methods for solids working within the framework of density-functional the- ory (DFT), to orbital-dependent functionals for the exchange-correlation (xc) energy. In con- trast to the commonly applied local-density approximation (LDA) and generalized gradient approximation (GGA) for the xc energy, orbital-dependent functionals depend directly on the Kohn-Sham (KS) orbitals and only indirectly on the density. Two dierent schemes that deal with orbital-dependent functionals, the KS and the gen- eralized Kohn-Sham (gKS) formalism, have been realized. While the KS scheme requires a local multiplicative xc potential, the gKS scheme allows for a non-local potential in the one- particle Schrödinger equations. Hybrid functionals, combining some amount of the orbital-dependent exact exchange en- ergy with local or semi-local functionals of the density, are implemented within the gKS scheme.
    [Show full text]
  • Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices∗
    PRECONDITIONERS FOR SYMMETRIZED TOEPLITZ AND MULTILEVEL TOEPLITZ MATRICES∗ J. PESTANAy Abstract. When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The precondi- tioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments. Key words. Toeplitz matrix, multilevel Toeplitz matrix, symmetrization, preconditioning, Krylov subspace method AMS subject classifications. 65F08, 65F10, 15B05, 35R11 1. Introduction. Linear systems (1.1) Anx = b; n×n n where An 2 R is a Toeplitz or multilevel Toeplitz matrix, and b 2 R arise in a range of applications. These include the discretization of partial differential and integral equations, time series analysis, and signal and image processing [7, 27]. Additionally, demand for fast numerical methods for fractional diffusion problems| which have recently received significant attention|has renewed interest in the solution of Toeplitz and Toeplitz-like systems [10, 26, 31, 32, 46]. Preconditioned iterative methods are often used to solve systems of the form (1.1). When An is Hermitian, CG [18] and MINRES [29] can be applied, and their descrip- tive convergence rate bounds guide the construction of effective preconditioners [7, 27]. On the other hand, convergence rates of preconditioned iterative methods for non- symmetric Toeplitz matrices are difficult to describe.
    [Show full text]
  • Linear Algebra Review
    Linear Programming Lecture 1: Linear Algebra Review Lecture 1: Linear Algebra Review Linear Programming 1 / 24 1 Linear Algebra Review 2 Linear Algebra Review 3 Block Structured Matrices 4 Gaussian Elimination Matrices 5 Gauss-Jordan Elimination (Pivoting) Lecture 1: Linear Algebra Review Linear Programming 2 / 24 Matrices in Rm×n A 2 Rm×n columns rows 2 3 2 3 a11 a12 ::: a1n a1• 6 a21 a22 ::: a2n 7 6 a2• 7 A = 6 7 = a a ::: a = 6 7 6 . .. 7 •1 •2 •n 6 . 7 4 . 5 4 . 5 am1 am2 ::: amn am• 2 3 2 T 3 a11 a21 ::: am1 a•1 T 6 a12 a22 ::: am2 7 6 a•2 7 AT = 6 7 = 6 7 = aT aT ::: aT 6 . .. 7 6 . 7 1• 2• m• 4 . 5 4 . 5 T a1n a2n ::: amn a•n Lecture 1: Linear Algebra Review Linear Programming 3 / 24 Matrix Vector Multiplication A column space view of matrix vector multiplication. 2 a11 a12 ::: a1n 3 2 x1 3 2 a11 3 2 a12 3 2 a1n 3 6 a21 a22 ::: a2n 7 6 x2 7 6 a21 7 6 a22 7 6 a2n 7 6 7 6 7 = x 6 7+ x 6 7+ ··· + x 6 7 6 . .. 7 6 . 7 1 6 . 7 2 6 . 7 n 6 . 7 4 . 5 4 . 5 4 . 5 4 . 5 4 . 5 am1 am2 ::: amn xn am1 am2 amn = x1 a•1 + x2 a•2 + ··· + xn a•n A linear combination of the columns. Lecture 1: Linear Algebra Review Linear Programming 4 / 24 The Range of a Matrix Let A 2 Rm×n (an m × n matrix having real entries).
    [Show full text]
  • Cluster Algebras
    Cluster Algebras Philipp Lampe December 4, 2013 2 Contents 1 Informal introduction 5 1.1 Sequences of Laurent polynomials . .5 1.2 Exercises . .7 2 What are cluster algebras? 9 2.1 Quivers and adjacency matrices . .9 2.2 Quiver mutation . 14 2.3 Cluster algebras attached to quivers . 24 2.4 Skew-symmetrizable matrices, ice quivers and cluster algebras . 29 2.5 Exercises . 31 3 Examples of cluster algebras 33 3.1 Sequences and Diophantine equations attached to cluster algebras . 33 3.2 The Kronecker cluster algebra . 40 3.3 Cluster algebras of type A . 44 3.4 Exercises . 50 4 The Laurent phenomenon 53 4.1 The proof of the Laurent phenomenon . 53 4.2 Exercises . 57 5 Solutions to exercises 59 Bibliography 63 3 4 CONTENTS Chapter 1 Informal introduction 1.1 Sequences of Laurent polynomials In the lecture we wish to give an introduction to Sergey Fomin and Andrei Zelevinsky’s theory of cluster algebras. Fomin and Zelevinsky have introduced and studied cluster algebras in a series of four influential articles [FZ, FZ2, BFZ3, FZ4] (one of which is coauthored with Arkady Berenstein). Although their initial motivation comes from Lie theory, the definition of a cluster algebra is very elementary. We will give the precise definition in Chapter 2, but to give the reader a first idea let a and b be two undeterminates and let us consider the map b + 1 F : (a, b) 7! b, . a Surprisingly, the non-trivial algebraic identity F5 = id holds. This equation has a long and colour- ful history.
    [Show full text]
  • RIMS-1650 Integrable Systems: the R-Matrix Approach by Michael Semenov-Tian-Shansky December 2008 RESEARCH INSTITUTE for MATHEMA
    RIMS-1650 Integrable Systems: the r-matrix Approach By Michael Semenov-Tian-Shansky December 2008 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan Integrable Systems: the r-matrix Approach Michael Semenov-Tian-Shansky Universite´ de Bourgogne UFR Sciences & Techniques 9 av- enue Alain Savary BP 47970 - 21078 Dijon Cedex FRANCE Current address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 JAPAN 1991 Mathematics Subject Classification. Primary 37K30 Secondary 17B37 17B68 37K10 Key words and phrases. Lie groups, Lie algebras, Poisson geometry, Poisson Lie groups, Symplectic geometry, Integrable systems, classical Yang-Baxter equation, Virasoro algebra, Differential Galois Theory Abstract. These lectures cover the theory of classical r-matrices sat- isfying the classical Yang–Baxter equations and their basic applications in the theory of Integrable Systems as well as in geometry of Poisson Lie groups. Special attention is given to the factorization problems associ- ated with classical r-matrices, the theory of dressing transformations, the geometric theory of difference zero curvature equations. We also discuss the relations between the Virasoro algebra and the Poisson Lie groups. Contents Introduction 4 Lecture 1. Preliminaries: Poisson Brackets, Poisson and Symplectic Manifolds, Symplectic Leaves, Reduction 7 1.1. Poisson Manifolds 7 1.2. Lie–Poisson Brackets 9 1.3. Poisson and Hamilton Reduction 12 1.4. Cotangent Bundle of a Lie Group. 18 Lecture 2. The r-Matrix Method and the Main Theorem 22 2.1. Introduction 22 2.2. Lie Dialgebras and Involutivity Theorem 23 2.3. Factorization Theorem 25 2.4. Factorization Theorem and Hamiltonian Reduction 26 2.5. Classical Yang-Baxter Identity and the General Theory of Classical r-Matrices 30 2.6.
    [Show full text]