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Perron–Frobenius Theory of Nonnegative Matrices
Buy from AMAZON.com http://www.amazon.com/exec/obidos/ASIN/0898714540 CHAPTER 8 Perron–Frobenius Theory of Nonnegative Matrices 8.1 INTRODUCTION m×n A ∈ is said to be a nonnegative matrix whenever each aij ≥ 0, and this is denoted by writing A ≥ 0. In general, A ≥ B means that each aij ≥ bij. Similarly, A is a positive matrix when each aij > 0, and this is denoted by writing A > 0. More generally, A > B means that each aij >bij. Applications abound with nonnegative and positive matrices. In fact, many of the applications considered in this text involve nonnegative matrices. For example, the connectivity matrix C in Example 3.5.2 (p. 100) is nonnegative. The discrete Laplacian L from Example 7.6.2 (p. 563) leads to a nonnegative matrix because (4I − L) ≥ 0. The matrix eAt that defines the solution of the system of differential equations in the mixing problem of Example 7.9.7 (p. 610) is nonnegative for all t ≥ 0. And the system of difference equations p(k)=Ap(k − 1) resulting from the shell game of Example 7.10.8 (p. 635) has Please report violations to [email protected] a nonnegativeCOPYRIGHTED coefficient matrix A. Since nonnegative matrices are pervasive, it’s natural to investigate their properties, and that’s the purpose of this chapter. A primary issue concerns It is illegal to print, duplicate, or distribute this material the extent to which the properties A > 0 or A ≥ 0 translate to spectral properties—e.g., to what extent does A have positive (or nonnegative) eigen- values and eigenvectors? The topic is called the “Perron–Frobenius theory” because it evolved from the contributions of the German mathematicians Oskar (or Oscar) Perron 89 and 89 Oskar Perron (1880–1975) originally set out to fulfill his father’s wishes to be in the family busi- Buy online from SIAM Copyright c 2000 SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 662 Chapter 8 Perron–Frobenius Theory of Nonnegative Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 Ferdinand Georg Frobenius. -
New Algorithms in the Frobenius Matrix Algebra for Polynomial Root-Finding
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports CUNY Academic Works 2014 TR-2014006: New Algorithms in the Frobenius Matrix Algebra for Polynomial Root-Finding Victor Y. Pan Ai-Long Zheng How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_cs_tr/397 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] New Algoirthms in the Frobenius Matrix Algebra for Polynomial Root-finding ∗ Victor Y. Pan[1,2],[a] and Ai-Long Zheng[2],[b] Supported by NSF Grant CCF-1116736 and PSC CUNY Award 64512–0042 [1] Department of Mathematics and Computer Science Lehman College of the City University of New York Bronx, NY 10468 USA [2] Ph.D. Programs in Mathematics and Computer Science The Graduate Center of the City University of New York New York, NY 10036 USA [a] [email protected] http://comet.lehman.cuny.edu/vpan/ [b] [email protected] Abstract In 1996 Cardinal applied fast algorithms in Frobenius matrix algebra to complex root-finding for univariate polynomials, but he resorted to some numerically unsafe techniques of symbolic manipulation with polynomials at the final stages of his algorithms. We extend his work to complete the computations by operating with matrices at the final stage as well and also to adjust them to real polynomial root-finding. Our analysis and experiments show efficiency of the resulting algorithms. 2000 Math. -
4.6 Matrices Dimensions: Number of Rows and Columns a Is a 2 X 3 Matrix
Matrices are rectangular arrangements of data that are used to represent information in tabular form. 1 0 4 A = 3 - 6 8 a23 4.6 Matrices Dimensions: number of rows and columns A is a 2 x 3 matrix Elements of a matrix A are denoted by aij. a23 = 8 1 2 Data about many kinds of problems can often be represented by Matrix of coefficients matrix. Solutions to many problems can be obtained by solving e.g Average temperatures in 3 different cities for each month: systems of linear equations. For example, the constraints of a problem are represented by the system of linear equations 23 26 38 47 58 71 78 77 69 55 39 33 x + y = 70 A = 14 21 33 38 44 57 61 59 49 38 25 21 3 cities 24x + 14y = 1180 35 46 54 67 78 86 91 94 89 75 62 51 12 months Jan - Dec 1 1 rd The matrix A = Average temp. in the 3 24 14 city in July, a37, is 91. is the matrix of coefficient for this system of linear equations. 3 4 In a matrix, the arrangement of the entries is significant. Square Matrix is a matrix in which the number of Therefore, for two matrices to be equal they must have rows equals the number of columns. the same dimensions and the same entries in each location. •Main Diagonal: in a n x n square matrix, the elements a , a , a , …, a form the main Example: Let 11 22 33 nn diagonal of the matrix. -
Recent Developments in Boolean Matrix Factorization
Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI-20) Survey Track Recent Developments in Boolean Matrix Factorization Pauli Miettinen1∗ and Stefan Neumann2y 1University of Eastern Finland, Kuopio, Finland 2University of Vienna, Vienna, Austria pauli.miettinen@uef.fi, [email protected] Abstract analysts, and van den Broeck and Darwiche [2013] to lifted inference community, to name but a few examples. Even more The goal of Boolean Matrix Factorization (BMF) is recently, Ravanbakhsh et al. [2016] studied the problem in to approximate a given binary matrix as the product the framework of machine learning, increasing the interest of of two low-rank binary factor matrices, where the that community, and Chandran et al. [2016]; Ban et al. [2019]; product of the factor matrices is computed under Fomin et al. [2019] (re-)launched the interest in the problem the Boolean algebra. While the problem is computa- in the theory community. tionally hard, it is also attractive because the binary The various fields in which BMF is used are connected by nature of the factor matrices makes them highly in- the desire to “keep the data binary”. This can be because terpretable. In the last decade, BMF has received a the factorization is used as a pre-processing step, and the considerable amount of attention in the data mining subsequent methods require binary input, or because binary and formal concept analysis communities and, more matrices are more interpretable in the application domain. In recently, the machine learning and the theory com- the latter case, Boolean algebra is often preferred over the field munities also started studying BMF. -
Subspace-Preserving Sparsification of Matrices with Minimal Perturbation
Subspace-preserving sparsification of matrices with minimal perturbation to the near null-space. Part I: Basics Chetan Jhurani Tech-X Corporation 5621 Arapahoe Ave Boulder, Colorado 80303, U.S.A. Abstract This is the first of two papers to describe a matrix sparsification algorithm that takes a general real or complex matrix as input and produces a sparse output matrix of the same size. The non-zero entries in the output are chosen to minimize changes to the singular values and singular vectors corresponding to the near null-space of the input. The output matrix is constrained to preserve left and right null-spaces exactly. The sparsity pattern of the output matrix is automatically determined or can be given as input. If the input matrix belongs to a common matrix subspace, we prove that the computed sparse matrix belongs to the same subspace. This works with- out imposing explicit constraints pertaining to the subspace. This property holds for the subspaces of Hermitian, complex-symmetric, Hamiltonian, cir- culant, centrosymmetric, and persymmetric matrices, and for each of the skew counterparts. Applications of our method include computation of reusable sparse pre- conditioning matrices for reliable and efficient solution of high-order finite element systems. The second paper in this series [1] describes our open- arXiv:1304.7049v1 [math.NA] 26 Apr 2013 source implementation, and presents further technical details. Keywords: Sparsification, Spectral equivalence, Matrix structure, Convex optimization, Moore-Penrose pseudoinverse Email address: [email protected] (Chetan Jhurani) Preprint submitted to Computers and Mathematics with Applications October 10, 2018 1. Introduction We present and analyze a matrix-valued optimization problem formulated to sparsify matrices while preserving the matrix null-spaces and certain spe- cial structural properties. -
Matrices and Linear Algebra
Chapter 26 Matrices and linear algebra ap,mat Contents 26.1 Matrix algebra........................................... 26.2 26.1.1 Determinant (s,mat,det)................................... 26.2 26.1.2 Eigenvalues and eigenvectors (s,mat,eig).......................... 26.2 26.1.3 Hermitian and symmetric matrices............................. 26.3 26.1.4 Matrix trace (s,mat,trace).................................. 26.3 26.1.5 Inversion formulas (s,mat,mil)............................... 26.3 26.1.6 Kronecker products (s,mat,kron).............................. 26.4 26.2 Positive-definite matrices (s,mat,pd) ................................ 26.4 26.2.1 Properties of positive-definite matrices........................... 26.5 26.2.2 Partial order......................................... 26.5 26.2.3 Diagonal dominance.................................... 26.5 26.2.4 Diagonal majorizers..................................... 26.5 26.2.5 Simultaneous diagonalization................................ 26.6 26.3 Vector norms (s,mat,vnorm) .................................... 26.6 26.3.1 Examples of vector norms.................................. 26.6 26.3.2 Inequalities......................................... 26.7 26.3.3 Properties.......................................... 26.7 26.4 Inner products (s,mat,inprod) ................................... 26.7 26.4.1 Examples.......................................... 26.7 26.4.2 Properties.......................................... 26.8 26.5 Matrix norms (s,mat,mnorm) .................................... 26.8 26.5.1 -
A Logarithmic Minimization Property of the Unitary Polar Factor in The
A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm Patrizio Neff∗ Yuji Nakatsukasa† Andreas Fischle‡ October 30, 2018 Abstract The unitary polar factor Q = Up in the polar decomposition of Z = Up H is the minimizer over unitary matrices Q for both Log(Q∗Z) 2 and its Hermitian part k k sym (Log(Q∗Z)) 2 over both R and C for any given invertible matrix Z Cn×n and k * k ∈ any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any n and for the Frobenius matrix norm in two and three dimensions. The result shows that the unitary polar factor is the nearest orthogonal matrix to Z not only in the normwise sense, but also in a geodesic distance. The derivation is based on Bhatia’s generalization of Bernstein’s trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all z C 0 ∈ \ { } −iϑ 2 2 −iϑ 2 2 min LogC(e z) = log z , min Re LogC(e z) = log z . ϑ∈(−π,π] | | | | || ϑ∈(−π,π] | | | | || Keywords: unitary polar factor, matrix logarithm, matrix exponential, Hermitian part, minimization, unitarily invariant norm, polar decomposition, sum of squared logarithms in- equality, optimality, matrix Lie-group, geodesic distance arXiv:1302.3235v4 [math.CA] 2 Jan 2014 AMS 2010 subject classification: 15A16, 15A18, 15A24, 15A44, 15A45, 26Dxx ∗Corresponding author, Head of Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Campus Essen, Thea-Leymann Str. -
A Complete Bibliography of Publications in Linear Algebra and Its Applications: 2010–2019
A Complete Bibliography of Publications in Linear Algebra and its Applications: 2010{2019 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 12 March 2021 Version 1.74 Title word cross-reference KY14, Rim12, Rud12, YHH12, vdH14]. 24 [KAAK11]. 2n − 3[BCS10,ˇ Hil13]. 2 × 2 [CGRVC13, CGSCZ10, CM14, DW11, DMS10, JK11, KJK13, MSvW12, Yan14]. (−1; 1) [AAFG12].´ (0; 1) 2 × 2 × 2 [Ber13b]. 3 [BBS12b, NP10, Ghe14a]. (2; 2; 0) [BZWL13, Bre14, CILL12, CKAC14, Fri12, [CI13, PH12]. (A; B) [PP13b]. (α, β) GOvdD14, GX12a, Kal13b, KK14, YHH12]. [HW11, HZM10]. (C; λ; µ)[dMR12].(`; m) p 3n2 − 2 2n3=2 − 3n [MR13]. [DFG10]. (H; m)[BOZ10].(κ, τ) p 3n2 − 2 2n3=23n [MR14a]. 3 × 3 [CSZ10, CR10c]. (λ, 2) [BBS12b]. (m; s; 0) [Dru14, GLZ14, Sev14]. 3 × 3 × 2 [Ber13b]. [GH13b]. (n − 3) [CGO10]. (n − 3; 2; 1) 3 × 3 × 3 [BH13b]. 4 [Ban13a, BDK11, [CCGR13]. (!) [CL12a]. (P; R)[KNS14]. BZ12b, CK13a, FP14, NSW13, Nor14]. 4 × 4 (R; S )[Tre12].−1 [LZG14]. 0 [AKZ13, σ [CJR11]. 5 Ano12-30, CGGS13, DLMZ14, Wu10a]. 1 [BH13b, CHY12, KRH14, Kol13, MW14a]. [Ano12-30, AHL+14, CGGS13, GM14, 5 × 5 [BAD09, DA10, Hil12a, Spe11]. 5 × n Kal13b, LM12, Wu10a]. 1=n [CNPP12]. [CJR11]. 6 1 <t<2 [Seo14]. 2 [AIS14, AM14, AKA13, [DK13c, DK11, DK12a, DK13b, Kar11a]. -
COMPUTING RELATIVELY LARGE ALGEBRAIC STRUCTURES by AUTOMATED THEORY EXPLORATION By
COMPUTING RELATIVELY LARGE ALGEBRAIC STRUCTURES BY AUTOMATED THEORY EXPLORATION by QURATUL-AIN MAHESAR A thesis submitted to The University of Birmingham for the degree of DOCTOR OF PHILOSOPHY School of Computer Science College of Engineering and Physical Sciences The University of Birmingham March 2014 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract Automated reasoning technology provides means for inference in a formal context via a multitude of disparate reasoning techniques. Combining different techniques not only increases the effectiveness of single systems but also provides a more powerful approach to solving hard problems. Consequently combined reasoning systems have been successfully employed to solve non-trivial mathematical problems in combinatorially rich domains that are intractable by traditional mathematical means. Nevertheless, the lack of domain specific knowledge often limits the effectiveness of these systems. In this thesis we investigate how the combination of diverse reasoning techniques can be employed to pre-compute additional knowledge to enable mathematical discovery in finite and potentially infinite domains that is otherwise not feasible. In particular, we demonstrate how we can exploit bespoke symbolic computations and automated theorem proving to automatically compute and evolve the structural knowledge of small size finite structures in the algebraic theory of quasigroups. -
View This Volume's Front and Back Matter
Combinatorics o f Nonnegative Matrice s This page intentionally left blank 10.1090/mmono/213 Translations o f MATHEMATICAL MONOGRAPHS Volume 21 3 Combinatorics o f Nonnegative Matrice s V. N. Sachko v V. E. Tarakano v Translated b y Valentin F . Kolchl n | yj | America n Mathematica l Societ y Providence, Rhod e Islan d EDITORIAL COMMITTE E AMS Subcommitte e Robert D . MacPherso n Grigorii A . Marguli s James D . Stashef f (Chair ) ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Freidli n (Chair ) B. H . Ca^KOB , B . E . TapaKaHO B KOMBMHATOPMKA HEOTPMUATEJIBHbl X MATPM U Hay^Hoe 143/iaTeJibCTB O TBE[ , MocKBa , 200 0 Translated fro m th e Russia n b y Dr . Valenti n F . Kolchi n 2000 Mathematics Subject Classification. Primar y 05-02 ; Secondary 05C50 , 15-02 , 15A48 , 93-02. Library o f Congress Cataloging-in-Publicatio n Dat a Sachkov, Vladimi r Nikolaevich . [Kombinatorika neotritsatel'nyk h matrits . English ] Combinatorics o f nonnegativ e matrice s / V . N . Sachkov , V . E . Tarakano v ; translate d b y Valentin F . Kolchin . p. cm . — (Translations o f mathematical monographs , ISS N 0065-928 2 ; v. 213) Includes bibliographica l reference s an d index. ISBN 0-8218-2788- X (acid-fre e paper ) 1. Non-negative matrices. 2 . Combinatorial analysis . I . Tarakanov, V . E. (Valerii Evgen'evich ) II. Title . III . Series. QA188.S1913 200 2 512.9'434—dc21 200207439 2 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapter fo r use in teachin g o r research . -
Perron–Frobenius Theorem for Matrices with Some Negative Entries
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 328 (2001) 57–68 www.elsevier.com/locate/laa Perron–Frobenius theorem for matrices with some negative entries Pablo Tarazaga a,∗,1, Marcos Raydan b,2, Ana Hurman c aDepartment of Mathematics, University of Puerto Rico, P.O. Box 5000, Mayaguez Campus, Mayaguez, PR 00681-5000, USA bDpto. de Computación, Facultad de Ciencias, Universidad Central de Venezuela, Ap. 47002, Caracas 1041-A, Venezuela cFacultad de Ingeniería y Facultad de Ciencias Económicas, Universidad Nacional del Cuyo, Mendoza, Argentina Received 17 December 1998; accepted 24 August 2000 Submitted by R.A. Brualdi Abstract We extend the classical Perron–Frobenius theorem to matrices with some negative entries. We study the cone of matrices that has the matrix of 1’s (eet) as the central ray, and such that any matrix inside the cone possesses a Perron–Frobenius eigenpair. We also find the maximum angle of any matrix with eet that guarantees this property. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Perron–Frobenius theorem; Cones of matrices; Optimality conditions 1. Introduction In 1907, Perron [4] made the fundamental discovery that the dominant eigenvalue of a matrix with all positive entries is positive and also that there exists a correspond- ing eigenvector with all positive entries. This result was later extended to nonnegative matrices by Frobenius [2] in 1912. In that case, there exists a nonnegative dominant ∗ Corresponding author. E-mail addresses: [email protected] (P. Tarazaga), [email protected] (M. -
Additional Details and Fortification for Chapter 1
APPENDIX A Additional Details and Fortification for Chapter 1 A.1 Matrix Classes and Special Matrices The matrices can be grouped into several classes based on their operational prop- erties. A short list of various classes of matrices is given in Tables A.1 and A.2. Some of these have already been described earlier, for example, elementary, sym- metric, hermitian, othogonal, unitary, positive definite/semidefinite, negative defi- nite/semidefinite, real, imaginary, and reducible/irreducible. Some of the matrix classes are defined based on the existence of associated matri- ces. For instance, A is a diagonalizable matrix if there exists nonsingular matrices T such that TAT −1 = D results in a diagonal matrix D. Connected with diagonalizable matrices are normal matrices. A matrix B is a normal matrix if BB∗ = B∗B.Normal matrices are guaranteed to be diagonalizable matrices. However, defective matri- ces are not diagonalizable. Once a matrix has been identified to be diagonalizable, then the following fact can be used for easier computation of integral powers of the matrix: A = T −1DT → Ak = T −1DT T −1DT ··· T −1DT = T −1DkT and then take advantage of the fact that ⎛ ⎞ dk 0 ⎜ 1 ⎟ K . D = ⎝ .. ⎠ K 0 dN Another set of related classes of matrices are the idempotent, projection, invo- lutory, nilpotent, and convergent matrices. These classes are based on the results of integral powers. Matrix A is idempotent if A2 = A, and if, in addition, A is hermitian, then A is known as a projection matrix. Projection matrices are used to partition an N-dimensional space into two subspaces that are orthogonal to each other.