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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]. -
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. -
Stat 5102 Notes: Regression
Stat 5102 Notes: Regression Charles J. Geyer April 27, 2007 In these notes we do not use the “upper case letter means random, lower case letter means nonrandom” convention. Lower case normal weight letters (like x and β) indicate scalars (real variables). Lowercase bold weight letters (like x and β) indicate vectors. Upper case bold weight letters (like X) indicate matrices. 1 The Model The general linear model has the form p X yi = βjxij + ei (1.1) j=1 where i indexes individuals and j indexes different predictor variables. Ex- plicit use of (1.1) makes theory impossibly messy. We rewrite it as a vector equation y = Xβ + e, (1.2) where y is a vector whose components are yi, where X is a matrix whose components are xij, where β is a vector whose components are βj, and where e is a vector whose components are ei. Note that y and e have dimension n, but β has dimension p. The matrix X is called the design matrix or model matrix and has dimension n × p. As always in regression theory, we treat the predictor variables as non- random. So X is a nonrandom matrix, β is a nonrandom vector of unknown parameters. The only random quantities in (1.2) are e and y. As always in regression theory the errors ei are independent and identi- cally distributed mean zero normal. This is written as a vector equation e ∼ Normal(0, σ2I), where σ2 is another unknown parameter (the error variance) and I is the identity matrix. This implies y ∼ Normal(µ, σ2I), 1 where µ = Xβ. -
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. -
DECOMPOSITION of SINGULAR MATRICES INTO IDEMPOTENTS 11 Us Show How to Construct Ai+1, Bi+1, Ci+1, Di+1
DECOMPOSITION OF SINGULAR MATRICES INTO IDEMPOTENTS ADEL ALAHMADI, S. K. JAIN, AND ANDRE LEROY Abstract. In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempo- tents and apply these factorizations for proving our main results. We generalize works due to Laffey ([12]) and Rao ([3]) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems (cf. [3], Theorems 5 and 7 and [4]). We also consider singular matrices over B´ezoutdomains as to when such a matrix is a product of idempotent matrices. 1. Introduction and definitions It was shown by Howie [10] that every mapping from a finite set X to itself with image of cardinality ≤ cardX − 1 is a product of idempotent mappings. Erd¨os[7] showed that every singular square matrix over a field can be expressed as a product of idempotent matrices and this was generalized by several authors to certain classes of rings, in particular, to division rings and euclidean domains [12]. Turning to singular elements let us mention two results: Rao [3] characterized, via continued fractions, singular matrices over a commutative PID that can be decomposed as a product of idempotent matrices and Hannah-O'Meara [9] showed, among other results, that for a right self-injective regular ring R, an element a is a product of idempotents if and only if Rr:ann(a) = l:ann(a)R= R(1 − a)R. The purpose of this paper is to provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and to apply these factorizations for proving our main results. -
Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 2, Pages 363{392 S 0894-0347(02)00412-5 Article electronically published on November 29, 2002 TOTALLY POSITIVE TOEPLITZ MATRICES AND QUANTUM COHOMOLOGY OF PARTIAL FLAG VARIETIES KONSTANZE RIETSCH 1. Introduction A matrix is called totally nonnegative if all of its minors are nonnegative. Totally nonnegative infinite Toeplitz matrices were studied first in the 1950's. They are characterized in the following theorem conjectured by Schoenberg and proved by Edrei. Theorem 1.1 ([10]). The Toeplitz matrix ∞×1 1 a1 1 0 1 a2 a1 1 B . .. C B . a2 a1 . C B C A = B .. .. .. C B ad . C B C B .. .. C Bad+1 . a1 1 C B C B . C B . .. .. a a .. C B 2 1 C B . C B .. .. .. .. ..C B C is totally nonnegative@ precisely if its generating function is of theA form, 2 (1 + βit) 1+a1t + a2t + =exp(tα) ; ··· (1 γit) i Y2N − where α R 0 and β1 β2 0,γ1 γ2 0 with βi + γi < . 2 ≥ ≥ ≥···≥ ≥ ≥···≥ 1 This beautiful result has been reproved many times; see [32]P for anP overview. It may be thought of as giving a parameterization of the totally nonnegative Toeplitz matrices by ~ N N ~ (α;(βi)i; (~γi)i) R 0 R 0 R 0 i(βi +~γi) < ; f 2 ≥ × ≥ × ≥ j 1g i X2N where β~i = βi βi+1 andγ ~i = γi γi+1. − − Received by the editors December 10, 2001 and, in revised form, September 14, 2002. 2000 Mathematics Subject Classification. Primary 20G20, 15A48, 14N35, 14N15. -
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. -
Mathematics Study Guide
Mathematics Study Guide Matthew Chesnes The London School of Economics September 28, 2001 1 Arithmetic of N-Tuples • Vectors specificed by direction and length. The length of a vector is called its magnitude p 2 2 or “norm.” For example, x = (x1, x2). Thus, the norm of x is: ||x|| = x1 + x2. pPn 2 • Generally for a vector, ~x = (x1, x2, x3, ..., xn), ||x|| = i=1 xi . • Vector Order: consider two vectors, ~x, ~y. Then, ~x> ~y iff xi ≥ yi ∀ i and xi > yi for some i. ~x>> ~y iff xi > yi ∀ i. • Convex Sets: A set is convex if whenever is contains x0 and x00, it also contains the line segment, (1 − α)x0 + αx00. 2 2 Vector Space Formulations in Economics • We expect a consumer to have a complete (preference) order over all consumption bundles x in his consumption set. If he prefers x to x0, we write, x x0. If he’s indifferent between x and x0, we write, x ∼ x0. Finally, if he weakly prefers x to x0, we write x x0. • The set X, {x ∈ X : x xˆ ∀ xˆ ∈ X}, is a convex set. It is all bundles of goods that make the consumer at least as well off as with his current bundle. 3 3 Complex Numbers • Define complex numbers as ordered pairs such that the first element in the vector is the real part of the number and the second is complex. Thus, the real number -1 is denoted by (-1,0). A complex number, 2+3i, can be expressed (2,3). • Define multiplication on the complex numbers as, Z · Z0 = (a, b) · (c, d) = (ac − bd, ad + bc). -
[Math.QA] 3 Dec 2001 Minors)
QUANTUM COHOMOLOGY RINGS OF GRASSMANNIANS AND TOTAL POSITIVITY KONSTANZE RIETSCH Abstract. We give a proof of a result of D. Peterson’s identifying the quan- tum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of GLn. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov–Witten invariants). Finally, we use the positiv- ity of these Gromov–Witten invariants to prove certain inequalities for Schur polynomials at roots of unity. 1. Introduction + The group U of unipotent upper–triangular matrices in GLn have on their coordinate ring a nice basis with positive structure constants. Namely one has the dual of the classical limit of Lusztig’s geometrically constructed canonical basis of the quantized enveloping algebra. The existence of this basis has been closely tied [10] to the ‘totally positive part’ of U + (the matrices with only nonnegative minors). + In this paper we study certain remarkable subvarieties Vd,n of U which come up in the stabilizer of a particular standard principal nilpotent element e as clo- sures of the 1–dimensional components under Bruhat decomposition. By a theorem of Dale Peterson’s [16] the quantum cohomology rings of Grassmannians may be identified with the coordinate rings of these varieties. Therefore like U + itself these varieties have coordinate rings with ‘canonical’ bases on them (this time coming from Schubert bases), and with positive structure constants.