Introduction to Cluster Algebras Chapters 1–3
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Parametrizations of K-Nonnegative Matrices
Parametrizations of k-Nonnegative Matrices Anna Brosowsky, Neeraja Kulkarni, Alex Mason, Joe Suk, Ewin Tang∗ October 2, 2017 Abstract Totally nonnegative (positive) matrices are matrices whose minors are all nonnegative (positive). We generalize the notion of total nonnegativity, as follows. A k-nonnegative (resp. k-positive) matrix has all minors of size k or less nonnegative (resp. positive). We give a generating set for the semigroup of k-nonnegative matrices, as well as relations for certain special cases, i.e. the k = n − 1 and k = n − 2 unitriangular cases. In the above two cases, we find that the set of k-nonnegative matrices can be partitioned into cells, analogous to the Bruhat cells of totally nonnegative matrices, based on their factorizations into generators. We will show that these cells, like the Bruhat cells, are homeomorphic to open balls, and we prove some results about the topological structure of the closure of these cells, and in fact, in the latter case, the cells form a Bruhat-like CW complex. We also give a family of minimal k-positivity tests which form sub-cluster algebras of the total positivity test cluster algebra. We describe ways to jump between these tests, and give an alternate description of some tests as double wiring diagrams. 1 Introduction A totally nonnegative (respectively totally positive) matrix is a matrix whose minors are all nonnegative (respectively positive). Total positivity and nonnegativity are well-studied phenomena and arise in areas such as planar networks, combinatorics, dynamics, statistics and probability. The study of total positivity and total nonnegativity admit many varied applications, some of which are explored in “Totally Nonnegative Matrices” by Fallat and Johnson [5]. -
CONSTRUCTING INTEGER MATRICES with INTEGER EIGENVALUES CHRISTOPHER TOWSE,∗ Scripps College
Applied Probability Trust (25 March 2016) CONSTRUCTING INTEGER MATRICES WITH INTEGER EIGENVALUES CHRISTOPHER TOWSE,∗ Scripps College ERIC CAMPBELL,∗∗ Pomona College Abstract In spite of the proveable rarity of integer matrices with integer eigenvalues, they are commonly used as examples in introductory courses. We present a quick method for constructing such matrices starting with a given set of eigenvectors. The main feature of the method is an added level of flexibility in the choice of allowable eigenvalues. The method is also applicable to non-diagonalizable matrices, when given a basis of generalized eigenvectors. We have produced an online web tool that implements these constructions. Keywords: Integer matrices, Integer eigenvalues 2010 Mathematics Subject Classification: Primary 15A36; 15A18 Secondary 11C20 In this paper we will look at the problem of constructing a good problem. Most linear algebra and introductory ordinary differential equations classes include the topic of diagonalizing matrices: given a square matrix, finding its eigenvalues and constructing a basis of eigenvectors. In the instructional setting of such classes, concrete “toy" examples are helpful and perhaps even necessary (at least for most students). The examples that are typically given to students are, of course, integer-entry matrices with integer eigenvalues. Sometimes the eigenvalues are repeated with multipicity, sometimes they are all distinct. Oftentimes, the number 0 is avoided as an eigenvalue due to the degenerate cases it produces, particularly when the matrix in question comes from a linear system of differential equations. Yet in [10], Martin and Wong show that “Almost all integer matrices have no integer eigenvalues," let alone all integer eigenvalues. -
Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition Jean-Guillaume Dumas, Clement Pernet, Ziad Sultan
Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition Jean-Guillaume Dumas, Clement Pernet, Ziad Sultan To cite this version: Jean-Guillaume Dumas, Clement Pernet, Ziad Sultan. Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition. Journal of Symbolic Computation, Elsevier, 2017, Special issue on ISSAC’15, 83, pp.187-210. 10.1016/j.jsc.2016.11.011. hal-01251223v1 HAL Id: hal-01251223 https://hal.archives-ouvertes.fr/hal-01251223v1 Submitted on 5 Jan 2016 (v1), last revised 14 May 2018 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition Jean-Guillaume Dumas Universit´eGrenoble Alpes, Laboratoire LJK, umr CNRS, BP53X, 51, av. des Math´ematiques, F38041 Grenoble, France Cl´ement Pernet Universit´eGrenoble Alpes, Laboratoire de l’Informatique du Parall´elisme, Universit´ede Lyon, France. Ziad Sultan Universit´eGrenoble Alpes, Laboratoire LJK and LIG, Inria, CNRS, Inovall´ee, 655, av. de l’Europe, F38334 St Ismier Cedex, France Abstract The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. -
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. -
Ratner's Work on Unipotent Flows and Impact
Ratner’s Work on Unipotent Flows and Its Impact Elon Lindenstrauss, Peter Sarnak, and Amie Wilkinson Dani above. As the name suggests, these theorems assert that the closures, as well as related features, of the orbits of such flows are very restricted (rigid). As such they provide a fundamental and powerful tool for problems connected with these flows. The brilliant techniques that Ratner in- troduced and developed in establishing this rigidity have been the blueprint for similar rigidity theorems that have been proved more recently in other contexts. We begin by describing the setup for the group of 푑×푑 matrices with real entries and determinant equal to 1 — that is, SL(푑, ℝ). An element 푔 ∈ SL(푑, ℝ) is unipotent if 푔−1 is a nilpotent matrix (we use 1 to denote the identity element in 퐺), and we will say a group 푈 < 퐺 is unipotent if every element of 푈 is unipotent. Connected unipotent subgroups of SL(푑, ℝ), in particular one-parameter unipo- Ratner presenting her rigidity theorems in a plenary tent subgroups, are basic objects in Ratner’s work. A unipo- address to the 1994 ICM, Zurich. tent group is said to be a one-parameter unipotent group if there is a surjective homomorphism defined by polyno- In this note we delve a bit more into Ratner’s rigidity theo- mials from the additive group of real numbers onto the rems for unipotent flows and highlight some of their strik- group; for instance ing applications, expanding on the outline presented by Elon Lindenstrauss is Alice Kusiel and Kurt Vorreuter professor of mathemat- 1 푡 푡2/2 1 푡 ics at The Hebrew University of Jerusalem. -
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. -
Integer Matrix Approximation and Data Mining
Noname manuscript No. (will be inserted by the editor) Integer Matrix Approximation and Data Mining Bo Dong · Matthew M. Lin · Haesun Park Received: date / Accepted: date Abstract Integer datasets frequently appear in many applications in science and en- gineering. To analyze these datasets, we consider an integer matrix approximation technique that can preserve the original dataset characteristics. Because integers are discrete in nature, to the best of our knowledge, no previously proposed technique de- veloped for real numbers can be successfully applied. In this study, we first conduct a thorough review of current algorithms that can solve integer least squares problems, and then we develop an alternative least square method based on an integer least squares estimation to obtain the integer approximation of the integer matrices. We discuss numerical applications for the approximation of randomly generated integer matrices as well as studies of association rule mining, cluster analysis, and pattern extraction. Our computed results suggest that our proposed method can calculate a more accurate solution for discrete datasets than other existing methods. Keywords Data mining · Matrix factorization · Integer least squares problem · Clustering · Association rule · Pattern extraction The first author's research was supported in part by the National Natural Science Foundation of China under grant 11101067 and the Fundamental Research Funds for the Central Universities. The second author's research was supported in part by the National Center for Theoretical Sciences of Taiwan and by the Ministry of Science and Technology of Taiwan under grants 104-2115-M-006-017-MY3 and 105-2634-E-002-001. The third author's research was supported in part by the Defense Advanced Research Projects Agency (DARPA) XDATA program grant FA8750-12-2-0309 and NSF grants CCF-0808863, IIS-1242304, and IIS-1231742. -
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 . -
Alternating Sign Matrices, Extensions and Related Cones
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/311671190 Alternating sign matrices, extensions and related cones Article in Advances in Applied Mathematics · May 2017 DOI: 10.1016/j.aam.2016.12.001 CITATIONS READS 0 29 2 authors: Richard A. Brualdi Geir Dahl University of Wisconsin–Madison University of Oslo 252 PUBLICATIONS 3,815 CITATIONS 102 PUBLICATIONS 1,032 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Combinatorial matrix theory; alternating sign matrices View project All content following this page was uploaded by Geir Dahl on 16 December 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Alternating sign matrices, extensions and related cones Richard A. Brualdi∗ Geir Dahly December 1, 2016 Abstract An alternating sign matrix, or ASM, is a (0; ±1)-matrix where the nonzero entries in each row and column alternate in sign, and where each row and column sum is 1. We study the convex cone generated by ASMs of order n, called the ASM cone, as well as several related cones and polytopes. Some decomposition results are shown, and we find a minimal Hilbert basis of the ASM cone. The notion of (±1)-doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. For instance, we give a new short proof of the linear characterization of the ASM polytope, in fact for a more general polytope. -
Tropical Totally Positive Matrices 3
TROPICAL TOTALLY POSITIVE MATRICES STEPHANE´ GAUBERT AND ADI NIV Abstract. We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields. Keywords: Total positivity; total nonnegativity; tropical geometry; compound matrix; permanent; Monge matrices; Grassmannian; Pl¨ucker coordinates. AMSC: 15A15 (Primary), 15A09, 15A18, 15A24, 15A29, 15A75, 15A80, 15B99. 1. Introduction 1.1. Motivation and background. A real matrix is said to be totally positive (resp. totally nonneg- ative) if all its minors are positive (resp. nonnegative). These matrices arise in several classical fields, such as oscillatory matrices (see e.g. [And87, §4]), or approximation theory (see e.g. [GM96]); they have appeared more recently in the theory of canonical bases for quantum groups [BFZ96]. We refer the reader to the monograph of Fallat and Johnson in [FJ11] or to the survey of Fomin and Zelevin- sky [FZ00] for more information. Totally positive/nonnegative matrices can be defined over any real closed field, and in particular, over nonarchimedean fields, like the field of Puiseux series with real coefficients. -
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 -
K-Theory of Furstenberg Transformation Group C ∗-Algebras
Canad. J. Math. Vol. 65 (6), 2013 pp. 1287–1319 http://dx.doi.org/10.4153/CJM-2013-022-x c Canadian Mathematical Society 2013 K-theory of Furstenberg Transformation Group C∗-algebras Kamran Reihani Abstract. This paper studies the K-theoretic invariants of the crossed product C∗-algebras associated n with an important family of homeomorphisms of the tori T called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2; C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between −n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdos),˝ a simple explicit formula for the asymptotic behavior of the sequence fang is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips. 1 Introduction Furstenberg transformations were introduced in [9] as the first examples of homeo- morphisms of the tori that under some necessary and sufficient conditions are mini- mal and uniquely ergodic.