The Symbiotic Relationship of Combinatorics and Matrix Theory
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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]. -
Quiz7 Problem 1
Quiz 7 Quiz7 Problem 1. Independence The Problem. In the parts below, cite which tests apply to decide on independence or dependence. Choose one test and show complete details. 1 ! 1 ! (a) Vectors ~v = ;~v = 1 −1 2 1 0 1 1 0 1 1 0 1 1 B C B C B C (b) Vectors ~v1 = @ −1 A ;~v2 = @ 1 A ;~v3 = @ 0 A 0 −1 −1 2 5 (c) Vectors ~v1;~v2;~v3 are data packages constructed from the equations y = x ; y = x ; y = x10 on (−∞; 1). (d) Vectors ~v1;~v2;~v3 are data packages constructed from the equations y = 1 + x; y = 1 − x; y = x3 on (−∞; 1). Basic Test. To show three vectors are independent, form the system of equations c1~v1 + c2~v2 + c3~v3 = ~0; then solve for c1; c2; c3. If the only solution possible is c1 = c2 = c3 = 0, then the vectors are independent. Linear Combination Test. A list of vectors is independent if and only if each vector in the list is not a linear combination of the remaining vectors, and each vector in the list is not zero. Subset Test. Any nonvoid subset of an independent set is independent. The basic test leads to three quick independence tests for column vectors. All tests use the augmented matrix A of the vectors, which for 3 vectors is A =< ~v1j~v2j~v3 >. Rank Test. The vectors are independent if and only if rank(A) equals the number of vectors. Determinant Test. Assume A is square. The vectors are independent if and only if the deter- minant of A is nonzero. -
Adjacency and Incidence Matrices
Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. -
A Brief Introduction to Spectral Graph Theory
A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY CATHERINE BABECKI, KEVIN LIU, AND OMID SADEGHI MATH 563, SPRING 2020 Abstract. There are several matrices that can be associated to a graph. Spectral graph theory is the study of the spectrum, or set of eigenvalues, of these matrices and its relation to properties of the graph. We introduce the primary matrices associated with graphs, and discuss some interesting questions that spectral graph theory can answer. We also discuss a few applications. 1. Introduction and Definitions This work is primarily based on [1]. We expect the reader is familiar with some basic graph theory and linear algebra. We begin with some preliminary definitions. Definition 1. Let Γ be a graph without multiple edges. The adjacency matrix of Γ is the matrix A indexed by V (Γ), where Axy = 1 when there is an edge from x to y, and Axy = 0 otherwise. This can be generalized to multigraphs, where Axy becomes the number of edges from x to y. Definition 2. Let Γ be an undirected graph without loops. The incidence matrix of Γ is the matrix M, with rows indexed by vertices and columns indexed by edges, where Mxe = 1 whenever vertex x is an endpoint of edge e. For a directed graph without loss, the directed incidence matrix N is defined by Nxe = −1; 1; 0 corresponding to when x is the head of e, tail of e, or not on e. Definition 3. Let Γ be an undirected graph without loops. The Laplace matrix of Γ is the matrix L indexed by V (G) with zero row sums, where Lxy = −Axy for x 6= y. -
Arxiv:1306.4805V3 [Math.OC] 6 Feb 2015 Used Greedy Techniques to Reorder Matrices
CONVEX RELAXATIONS FOR PERMUTATION PROBLEMS FAJWEL FOGEL, RODOLPHE JENATTON, FRANCIS BACH, AND ALEXANDRE D’ASPREMONT ABSTRACT. Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We write seri- ation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we derive several convex relax- ations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution, hence solve semi-supervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data. 1. INTRODUCTION We study optimization problems written over the set of permutations. While the relaxation techniques discussed in what follows are applicable to a much more general setting, most of the paper is centered on the seriation problem: we are given a similarity matrix between a set of n variables and assume that the variables can be ordered along a chain, where the similarity between variables decreases with their distance within this chain. The seriation problem seeks to reconstruct this linear ordering based on unsorted, possibly noisy, pairwise similarity information. This problem has its roots in archeology [Robinson, 1951] and also has direct applications in e.g. -
Crystals and Total Positivity on Orientable Surfaces 3
CRYSTALS AND TOTAL POSITIVITY ON ORIENTABLE SURFACES THOMAS LAM AND PAVLO PYLYAVSKYY Abstract. We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara-Noumi- Yamada [KNY] and an observation of Berenstein-Kazhdan [BK07b]. We show that our model on a cylinder gives a decomposition and parametrization of the totally nonnegative part of the rational unipotent loop group of GLn. Contents 1. Introduction 3 1.1. Networks on orientable surfaces 3 1.2. Factorizations and parametrizations of totally positive matrices 4 1.3. Crystals and networks 5 1.4. Measurements and moves 6 1.5. Symmetric functions and loop symmetric functions 7 1.6. Comparison of examples 7 Part 1. Boundary measurements on oriented surfaces 9 2. Networks and measurements 9 2.1. Oriented networks on surfaces 9 2.2. Polygon representation of oriented surfaces 9 2.3. Highway paths and cycles 10 arXiv:1008.1949v1 [math.CO] 11 Aug 2010 2.4. Boundary and cycle measurements 10 2.5. Torus with one vertex 11 2.6. Flows and intersection products in homology 12 2.7. Polynomiality 14 2.8. Rationality 15 2.9. -
Doubly Stochastic Matrices Whose Powers Eventually Stop
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 330 (2001) 25–30 www.elsevier.com/locate/laa Doubly stochastic matrices whose powers eventually stopୋ Suk-Geun Hwang a,∗, Sung-Soo Pyo b aDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang, South Korea Received 22 June 2000; accepted 14 November 2000 Submitted by R.A. Brualdi Abstract In this note we characterize doubly stochastic matrices A whose powers A, A2,A3,... + eventually stop, i.e., Ap = Ap 1 =···for some positive integer p. The characterization en- ables us to determine the set of all such matrices. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A51 Keywords: Doubly stochastic matrix; J-potent 1. Introduction Let R denote the real field. For positive integers m, n,letRm×n denote the set of all m × n matrices with real entries. As usual let Rn denote the set Rn×1.We call the members of Rn the n-vectors. The n-vector of 1’s is denoted by e,andthe identity matrix of order n is denoted by In. For two matrices A, B of the same size, let A B denote that all the entries of A − B are nonnegative. A matrix A is called nonnegative if A O. A nonnegative square matrix is called a doubly stochastic matrix if all of its row sums and column sums equal 1. -
Incidence Matrices and Interval Graphs
Pacific Journal of Mathematics INCIDENCE MATRICES AND INTERVAL GRAPHS DELBERT RAY FULKERSON AND OLIVER GROSS Vol. 15, No. 3 November 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 3, 1965 INCIDENCE MATRICES AND INTERVAL GRAPHS D. R. FULKERSON AND 0. A. GROSS According to present genetic theory, the fine structure of genes consists of linearly ordered elements. A mutant gene is obtained by alteration of some connected portion of this structure. By examining data obtained from suitable experi- ments, it can be determined whether or not the blemished portions of two mutant genes intersect or not, and thus inter- section data for a large number of mutants can be represented as an undirected graph. If this graph is an "interval graph," then the observed data is consistent with a linear model of the gene. The problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, l)-matrices: When can the rows of such a matrix be per- muted so as to make the l's in each column appear consecu- tively? A complete theory is obtained for this latter problem, culminating in a decomposition theorem which leads to a rapid algorithm for deciding the question, and for constructing the desired permutation when one exists. Let A — (dij) be an m by n matrix whose entries ai3 are all either 0 or 1. The matrix A may be regarded as the incidence matrix of elements el9 e2, , em vs. sets Sl9 S2, , Sn; that is, ai3 = 0 or 1 ac- cording as et is not or is a member of S3 . -
A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality
Electrical and Computer Engineering Publications Electrical and Computer Engineering 6-15-2017 A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality Songtao Lu Iowa State University, [email protected] Mingyi Hong Iowa State University Zhengdao Wang Iowa State University, [email protected] Follow this and additional works at: https://lib.dr.iastate.edu/ece_pubs Part of the Industrial Technology Commons, and the Signal Processing Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ ece_pubs/162. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Electrical and Computer Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Electrical and Computer Engineering Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality Abstract Symmetric nonnegative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection, and image segmentation. In this paper, we propose a novel nonconvex variable splitting method for solving SymNMF. The proposed algorithm is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points of the nonconvex SymNMF problem. Furthermore, it achieves a global sublinear convergence rate. We also show that the algorithm can be efficiently implemented in parallel. Further, sufficient conditions ear provided that guarantee the global and local optimality of the obtained solutions. -
Minimal Rank Decoupling of Full-Lattice CMV Operators With
MINIMAL RANK DECOUPLING OF FULL-LATTICE CMV OPERATORS WITH SCALAR- AND MATRIX-VALUED VERBLUNSKY COEFFICIENTS STEPHEN CLARK, FRITZ GESZTESY, AND MAXIM ZINCHENKO Abstract. Relations between half- and full-lattice CMV operators with scalar- and matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV oper- ators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients results in a perturbation of twice the minimal rank. The explicit form for the minimal rank perturbation and the resulting two half-lattice CMV matrices are obtained. In addition, formulas relating the Weyl–Titchmarsh m-functions (resp., matrices) associated with the involved CMV operators and their Green’s functions (resp., matrices) are derived. 1. Introduction CMV operators are a special class of unitary semi-infinite or doubly-infinite five-diagonal matrices which received enormous attention in recent years. We refer to (2.8) and (3.18) for the explicit form of doubly infinite CMV operators on Z in the case of scalar, respectively, matrix-valued Verblunsky coefficients. For the corresponding half-lattice CMV operators we refer to (2.16) and (3.26). The actual history of CMV operators (with scalar Verblunsky coefficients) is somewhat intrigu- ing: The corresponding unitary semi-infinite five-diagonal matrices were first introduced in 1991 by Bunse–Gerstner and Elsner [15], and subsequently discussed in detail by Watkins [82] in 1993 (cf. the discussion in Simon [73]). They were subsequently rediscovered by Cantero, Moral, and Vel´azquez (CMV) in [17]. -
Random Matrices, Magic Squares and Matching Polynomials
Random Matrices, Magic Squares and Matching Polynomials Persi Diaconis Alex Gamburd∗ Departments of Mathematics and Statistics Department of Mathematics Stanford University, Stanford, CA 94305 Stanford University, Stanford, CA 94305 [email protected] [email protected] Submitted: Jul 22, 2003; Accepted: Dec 23, 2003; Published: Jun 3, 2004 MR Subject Classifications: 05A15, 05E05, 05E10, 05E35, 11M06, 15A52, 60B11, 60B15 Dedicated to Richard Stanley on the occasion of his 60th birthday Abstract Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zeta- function, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed. 1 Introduction Two noteworthy developments took place recently in Random Matrix Theory. One is the discovery and exploitation of the connections between eigenvalue statistics and the longest- increasing subsequence problem in enumerative combinatorics [1, 4, 5, 47, 59]; another is the outburst of interest in characteristic polynomials of Random Matrices and associated global statistics, particularly in connection with the moments of the Riemann zeta function and other L-functions [41, 14, 35, 36, 15, 16]. The purpose of this paper is to point out some connections between the distribution of the coefficients of characteristic polynomials of random matrices and some classical problems in enumerative combinatorics. -
Z Matrices, Linear Transformations, and Tensors
Z matrices, linear transformations, and tensors M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland, USA [email protected] *************** International Conference on Tensors, Matrices, and their Applications Tianjin, China May 21-24, 2016 Z matrices, linear transformations, and tensors – p. 1/35 This is an expository talk on Z matrices, transformations on proper cones, and tensors. The objective is to show that these have very similar properties. Z matrices, linear transformations, and tensors – p. 2/35 Outline • The Z-property • M and strong (nonsingular) M-properties • The P -property • Complementarity problems • Zero-sum games • Dynamical systems Z matrices, linear transformations, and tensors – p. 3/35 Some notation • Rn : The Euclidean n-space of column vectors. n n • R+: Nonnegative orthant, x ∈ R+ ⇔ x ≥ 0. n n n • R++ : The interior of R+, x ∈++⇔ x > 0. • hx,yi: Usual inner product between x and y. • Rn×n: The space of all n × n real matrices. • σ(A): The set of all eigenvalues of A ∈ Rn×n. Z matrices, linear transformations, and tensors – p. 4/35 The Z-property A =[aij] is an n × n real matrix • A is a Z-matrix if aij ≤ 0 for all i =6 j. (In economics literature, −A is a Metzler matrix.) • We can write A = rI − B, where r ∈ R and B ≥ 0. Let ρ(B) denote the spectral radius of B. • A is an M-matrix if r ≥ ρ(B), • nonsingular (strong) M-matrix if r > ρ(B). Z matrices, linear transformations, and tensors – p. 5/35 The P -property • A is a P -matrix if all its principal minors are positive.