Totally Nonnegative Matrices
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Eigenvalues and Eigenvectors of Tridiagonal Matrices∗
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 15, pp. 115-133, April 2006 ELA http://math.technion.ac.il/iic/ela EIGENVALUES AND EIGENVECTORS OF TRIDIAGONAL MATRICES∗ SAID KOUACHI† Abstract. This paper is continuation of previous work by the present author, where explicit formulas for the eigenvalues associated with several tridiagonal matrices were given. In this paper the associated eigenvectors are calculated explicitly. As a consequence, a result obtained by Wen- Chyuan Yueh and independently by S. Kouachi, concerning the eigenvalues and in particular the corresponding eigenvectors of tridiagonal matrices, is generalized. Expressions for the eigenvectors are obtained that differ completely from those obtained by Yueh. The techniques used herein are based on theory of recurrent sequences. The entries situated on each of the secondary diagonals are not necessary equal as was the case considered by Yueh. Key words. Eigenvectors, Tridiagonal matrices. AMS subject classifications. 15A18. 1. Introduction. The subject of this paper is diagonalization of tridiagonal matrices. We generalize a result obtained in [5] concerning the eigenvalues and the corresponding eigenvectors of several tridiagonal matrices. We consider tridiagonal matrices of the form −α + bc1 00 ... 0 a1 bc2 0 ... 0 .. .. 0 a2 b . An = , (1) .. .. .. 00. 0 . .. .. .. . cn−1 0 ... ... 0 an−1 −β + b n−1 n−1 ∞ where {aj}j=1 and {cj}j=1 are two finite subsequences of the sequences {aj}j=1 and ∞ {cj}j=1 of the field of complex numbers C, respectively, and α, β and b are complex numbers. We suppose that 2 d1, if j is odd ajcj = 2 j =1, 2, ..., (2) d2, if j is even where d 1 and d2 are complex numbers. -
18.06 Linear Algebra, Problem Set 2 Solutions
18.06 Problem Set 2 Solution Total: 100 points Section 2.5. Problem 24: Use Gauss-Jordan elimination on [U I] to find the upper triangular −1 U : 2 3 2 3 2 3 1 a b 1 0 0 −1 4 5 4 5 4 5 UU = I 0 1 c x1 x2 x3 = 0 1 0 : 0 0 1 0 0 1 −1 Solution (4 points): Row reduce [U I] to get [I U ] as follows (here Ri stands for the ith row): 2 3 2 3 1 a b 1 0 0 (R1 = R1 − aR2) 1 0 b − ac 1 −a 0 4 5 4 5 0 1 c 0 1 0 −! (R2 = R2 − cR2) 0 1 0 0 1 −c 0 0 1 0 0 1 0 0 1 0 0 1 ( ) 2 3 R1 = R1 − (b − ac)R3 1 0 0 1 −a ac − b −! 40 1 0 0 1 −c 5 : 0 0 1 0 0 1 Section 2.5. Problem 40: (Recommended) A is a 4 by 4 matrix with 1's on the diagonal and −1 −a; −b; −c on the diagonal above. Find A for this bidiagonal matrix. −1 Solution (12 points): Row reduce [A I] to get [I A ] as follows (here Ri stands for the ith row): 2 3 1 −a 0 0 1 0 0 0 6 7 60 1 −b 0 0 1 0 07 4 5 0 0 1 −c 0 0 1 0 0 0 0 1 0 0 0 1 2 3 (R1 = R1 + aR2) 1 0 −ab 0 1 a 0 0 6 7 (R2 = R2 + bR2) 60 1 0 −bc 0 1 b 07 −! 4 5 (R3 = R3 + cR4) 0 0 1 0 0 0 1 c 0 0 0 1 0 0 0 1 2 3 (R1 = R1 + abR3) 1 0 0 0 1 a ab abc (R = R + bcR ) 60 1 0 0 0 1 b bc 7 −! 2 2 4 6 7 : 40 0 1 0 0 0 1 c 5 0 0 0 1 0 0 0 1 Alternatively, write A = I − N. -
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]. -
[Math.RA] 19 Jun 2003 Two Linear Transformations Each Tridiagonal with Respect to an Eigenbasis of the Ot
Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array∗ Paul Terwilliger Abstract Let K denote a field. Let d denote a nonnegative integer and consider a sequence ∗ K p = (θi,θi , i = 0...d; ϕj , φj, j = 1...d) consisting of scalars taken from . We call p ∗ ∗ a parameter array whenever: (PA1) θi 6= θj, θi 6= θj if i 6= j, (0 ≤ i, j ≤ d); (PA2) i−1 θh−θd−h ∗ ∗ ϕi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) ϕi = φ1 + (θ − θ )(θi−1 − θ ) h=0 θ0−θd i 0 d i−1 θh−θd−h ∗ ∗ (1 ≤ i ≤ d); (PA4) φi = ϕ1 + (Pθ − θ )(θd−i+1 − θ0) (1 ≤ i ≤ d); h=0 θ0−θd i 0 −1 ∗ ∗ ∗ ∗ −1 (PA5) (θi−2 − θi+1)(θi−1 − θi) P, (θi−2 − θi+1)(θi−1 − θi ) are equal and independent of i for 2 ≤ i ≤ d − 1. In [13] we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1, PA2. Let ∗ ∗ A, B, A ,B denote the matrices in Matd+1(K) which have entries Aii = θi, Bii = θd−i, ∗ ∗ ∗ ∗ ∗ ∗ Aii = θi , Bii = θi (0 ≤ i ≤ d), Ai,i−1 = 1, Bi,i−1 = 1, Ai−1,i = ϕi, Bi−1,i = φi (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies −1 PA3–PA5; (ii) there exists an invertible G ∈ Matd+1(K) such that G AG = B and G−1A∗G = B∗; (iii) for 0 ≤ i ≤ d the polynomial i ∗ ∗ ∗ ∗ ∗ ∗ (λ − θ0)(λ − θ1) · · · (λ − θn−1)(θi − θ0)(θi − θ1) · · · (θi − θn−1) ϕ1ϕ2 · · · ϕ nX=0 n is a scalar multiple of the polynomial i ∗ ∗ ∗ ∗ ∗ ∗ (λ − θd)(λ − θd−1) · · · (λ − θd−n+1)(θ − θ )(θ − θ ) · · · (θ − θ ) i 0 i 1 i n−1 . -
Self-Interlacing Polynomials Ii: Matrices with Self-Interlacing Spectrum
SELF-INTERLACING POLYNOMIALS II: MATRICES WITH SELF-INTERLACING SPECTRUM MIKHAIL TYAGLOV Abstract. An n × n matrix is said to have a self-interlacing spectrum if its eigenvalues λk, k = 1; : : : ; n, are distributed as follows n−1 λ1 > −λ2 > λ3 > ··· > (−1) λn > 0: A method for constructing sign definite matrices with self-interlacing spectra from totally nonnegative ones is presented. We apply this method to bidiagonal and tridiagonal matrices. In particular, we generalize a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries. 1. Introduction In [5] there were introduced the so-called self-interlacing polynomials. A polynomial p(z) is called self- interlacing if all its roots are real, semple and interlacing the roots of the polynomial p(−z). It is easy to see that if λk, k = 1; : : : ; n, are the roots of a self-interlacing polynomial, then the are distributed as follows n−1 (1.1) λ1 > −λ2 > λ3 > ··· > (−1) λn > 0; or n (1.2) − λ1 > λ2 > −λ3 > ··· > (−1) λn > 0: The polynomials whose roots are distributed as in (1.1) (resp. in (1.2)) are called self-interlacing of kind I (resp. of kind II). It is clear that a polynomial p(z) is self-interlacing of kind I if, and only if, the polynomial p(−z) is self-interlacing of kind II. Thus, it is enough to study self-interlacing of kind I, since all the results for self-interlacing of kind II will be obtained automatically. Definition 1.1. An n × n matrix is said to possess a self-interlacing spectrum if its eigenvalues λk, k = 1; : : : ; n, are real, simple, are distributed as in (1.1). -
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. -
QR Decomposition: History and Its Applications
Mathematics & Statistics Auburn University, Alabama, USA QR History Dec 17, 2010 Asymptotic result QR iteration QR decomposition: History and its EE Applications Home Page Title Page Tin-Yau Tam èèèUUUÎÎÎ JJ II J I Page 1 of 37 Æâ§w Go Back fÆêÆÆÆ Full Screen Close email: [email protected] Website: www.auburn.edu/∼tamtiny Quit 1. QR decomposition Recall the QR decomposition of A ∈ GLn(C): QR A = QR History Asymptotic result QR iteration where Q ∈ GLn(C) is unitary and R ∈ GLn(C) is upper ∆ with positive EE diagonal entries. Such decomposition is unique. Set Home Page a(A) := diag (r11, . , rnn) Title Page where A is written in column form JJ II J I A = (a1| · · · |an) Page 2 of 37 Go Back Geometric interpretation of a(A): Full Screen rii is the distance (w.r.t. 2-norm) between ai and span {a1, . , ai−1}, Close i = 2, . , n. Quit Example: 12 −51 4 6/7 −69/175 −58/175 14 21 −14 6 167 −68 = 3/7 158/175 6/175 0 175 −70 . QR −4 24 −41 −2/7 6/35 −33/35 0 0 35 History Asymptotic result QR iteration EE • QR decomposition is the matrix version of the Gram-Schmidt orthonor- Home Page malization process. Title Page JJ II • QR decomposition can be extended to rectangular matrices, i.e., if A ∈ J I m×n with m ≥ n (tall matrix) and full rank, then C Page 3 of 37 A = QR Go Back Full Screen where Q ∈ Cm×n has orthonormal columns and R ∈ Cn×n is upper ∆ Close with positive “diagonal” entries. -
The Geometry of Convex Cones Associated with the Lyapunov Inequality and the Common Lyapunov Function Problem∗
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 12, pp. 42-63, March 2005 ELA www.math.technion.ac.il/iic/ela THE GEOMETRY OF CONVEX CONES ASSOCIATED WITH THE LYAPUNOV INEQUALITY AND THE COMMON LYAPUNOV FUNCTION PROBLEM∗ OLIVER MASON† AND ROBERT SHORTEN‡ Abstract. In this paper, the structure of several convex cones that arise in the studyof Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence. Key words. Lyapunov functions and stability, Convex cones, Matrix equations. AMS subject classifications. 37B25, 47L07, 39B42. 1. Introduction and motivation. Recently, there has been considerable inter- est across the mathematics, computer science, and control engineering communities in the analysis and design of so-called hybrid dynamical systems [7,15,19,22,23]. Roughly speaking, a hybrid system is one whose behaviour can be described math- ematically by combining classical differential/difference equations with some logic based switching mechanism or rule. These systems arise in a wide variety of engineer- ing applications with examples occurring in the aircraft, automotive and communi- cations industries. In spite of the attention that hybrid systems have received in the recent past, important aspects of their behaviour are not yet completely understood. -
Chapter Four Determinants
Chapter Four Determinants In the first chapter of this book we considered linear systems and we picked out the special case of systems with the same number of equations as unknowns, those of the form T~x = ~b where T is a square matrix. We noted a distinction between two classes of T ’s. While such systems may have a unique solution or no solutions or infinitely many solutions, if a particular T is associated with a unique solution in any system, such as the homogeneous system ~b = ~0, then T is associated with a unique solution for every ~b. We call such a matrix of coefficients ‘nonsingular’. The other kind of T , where every linear system for which it is the matrix of coefficients has either no solution or infinitely many solutions, we call ‘singular’. Through the second and third chapters the value of this distinction has been a theme. For instance, we now know that nonsingularity of an n£n matrix T is equivalent to each of these: ² a system T~x = ~b has a solution, and that solution is unique; ² Gauss-Jordan reduction of T yields an identity matrix; ² the rows of T form a linearly independent set; ² the columns of T form a basis for Rn; ² any map that T represents is an isomorphism; ² an inverse matrix T ¡1 exists. So when we look at a particular square matrix, the question of whether it is nonsingular is one of the first things that we ask. This chapter develops a formula to determine this. (Since we will restrict the discussion to square matrices, in this chapter we will usually simply say ‘matrix’ in place of ‘square matrix’.) More precisely, we will develop infinitely many formulas, one for 1£1 ma- trices, one for 2£2 matrices, etc. -
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. -
Package 'Igraphmatch'
Package ‘iGraphMatch’ January 27, 2021 Type Package Title Tools Graph Matching Version 1.0.1 Description Graph matching methods and analysis. The package works for both 'igraph' objects and matrix objects. You provide the adjacency matrices of two graphs and some other information you might know, choose the graph matching method, and it returns the graph matching results. 'iGraphMatch' also provides a bunch of useful functions you might need related to graph matching. Depends R (>= 3.3.1) Imports clue (>= 0.3-54), Matrix (>= 1.2-11), igraph (>= 1.1.2), irlba, methods, Rcpp Suggests dplyr (>= 0.5.0), testthat (>= 2.0.0), knitr, rmarkdown VignetteBuilder knitr License GPL (>= 2) LazyData TRUE RoxygenNote 7.1.1 Language en-US Encoding UTF-8 LinkingTo Rcpp NeedsCompilation yes Author Daniel Sussman [aut, cre], Zihuan Qiao [aut], Joshua Agterberg [ctb], Lujia Wang [ctb], Vince Lyzinski [ctb] Maintainer Daniel Sussman <[email protected]> Repository CRAN Date/Publication 2021-01-27 16:00:02 UTC 1 2 bari_start R topics documented: bari_start . .2 best_matches . .4 C.Elegans . .5 center_graph . .6 check_seeds . .7 do_lap . .8 Enron . .9 get_perm . .9 graph_match_convex . 10 graph_match_ExpandWhenStuck . 12 graph_match_IsoRank . 13 graph_match_Umeyama . 15 init_start . 16 innerproduct . 17 lapjv . 18 lapmod . 18 largest_common_cc . 19 matched_adjs . 20 match_plot_igraph . 20 match_report . 22 pad.............................................. 23 row_cor . 24 rperm . 25 sample_correlated_gnp_pair . 25 sample_correlated_ieg_pair . 26 sample_correlated_sbm_pair . 28 split_igraph . 29 splrMatrix-class . 30 splr_sparse_plus_constant . 31 splr_to_sparse . 31 Index 32 bari_start Start matrix initialization Description initialize the start matrix for graph matching iteration. bari_start 3 Usage bari_start(nns, ns = 0, soft_seeds = NULL) rds_sinkhorn_start(nns, ns = 0, soft_seeds = NULL, distribution = "runif") rds_perm_bari_start(nns, ns = 0, soft_seeds = NULL, g = 1, is_splr = TRUE) rds_from_sim_start(nns, ns = 0, soft_seeds = NULL, sim) Arguments nns An integer. -
Section 2.4–2.5 Partitioned Matrices and LU Factorization
Section 2.4{2.5 Partitioned Matrices and LU Factorization Gexin Yu [email protected] College of William and Mary Gexin Yu [email protected] Section 2.4{2.5 Partitioned Matrices and LU Factorization One approach to simplify the computation is to partition a matrix into blocks. 2 3 0 −1 5 9 −2 3 Ex: A = 4 −5 2 4 0 −3 1 5. −8 −6 3 1 7 −4 This partition can also be written as the following 2 × 3 block matrix: A A A A = 11 12 13 A21 A22 A23 3 0 −1 In the block form, we have blocks A = and so on. 11 −5 2 4 partition matrices into blocks In real world problems, systems can have huge numbers of equations and un-knowns. Standard computation techniques are inefficient in such cases, so we need to develop techniques which exploit the internal structure of the matrices. In most cases, the matrices of interest have lots of zeros. Gexin Yu [email protected] Section 2.4{2.5 Partitioned Matrices and LU Factorization 2 3 0 −1 5 9 −2 3 Ex: A = 4 −5 2 4 0 −3 1 5. −8 −6 3 1 7 −4 This partition can also be written as the following 2 × 3 block matrix: A A A A = 11 12 13 A21 A22 A23 3 0 −1 In the block form, we have blocks A = and so on. 11 −5 2 4 partition matrices into blocks In real world problems, systems can have huge numbers of equations and un-knowns.