More Linear Algebra1

Total Page:16

File Type:pdf, Size:1020Kb

More Linear Algebra1 More Linear Algebra1 Chris Godsil Combinatorics & Optimization University of Waterloo ©2019 1version: January 29, 2021 ii Preface These notes are meant to provide an introduction to fundamental parts of lin- ear algebra, as might be applied to problems in combinatorics. I assume the reader has had a first course in linear algebra, and is familiar with determinants. iii To Do 1. Interlacing, via Courant-Fischer and by rational functions. Need equi- table partitions for this. 2. Walk modules. Controllable graphs. 3. Quadrics. 4. Matrix exponential. 5. Lie algebras, sl(2) repns. 6. Perron-Frobenius, symbolic dynamics. 7. Perturbation theory. iv Contents 1 Spaces and Subspaces 1 1.1 Vector Spaces . 1 1.2 Subspaces . 3 1.3 Linear Mappings . 4 1.4 Duals and Adjoints . 6 1.5 Bilinear Forms . 8 1.6 Counting . 9 1.7 Normal Forms . 11 1.8 Groebner Bases . 13 1.9 Codes . 15 2 Primary Decomposition 19 2.1 Modules . 19 2.2 Control Theory . 20 2.3 Sums .................................... 22 2.4 Invariant Sums . 23 2.5 Minimal Polynomials . 25 2.6 Primary Decomposition . 27 2.7 The Degree of the Minimal Polynomial . 29 2.8 Root Spaces . 30 2.9 Examples of Root Spaces . 31 2.10 Differential Equations . 33 2.11 Linear Recurrence Equations . 35 2.12 Diagonalizability . 36 3 Frobenius Normal Form 39 3.1 Companion Matrices . 39 3.2 Transposes . 41 v vi CONTENTS 3.3 Eigenvectors for Companion Matrices . 42 3.4 Inverses of Companion Matrices . 44 3.5 Cycles.................................... 46 3.6 Circulants and Cyclic Codes . 47 3.7 Frobenius Normal Form . 49 3.8 Applications . 52 3.9 Nilpotent Matrices . 53 3.10 A Similarity Condition . 55 3.11 Triangular Maps . 57 3.12 Triangulations . 59 3.13 The “Fundamental” “Theorem of Algebra” . 59 3.14 The Kronecker Product . 63 4 Orthogonality 67 4.1 Properties of Projections . 67 4.2 Matrices Representing Projections . 68 4.3 Least Squares . 70 4.4 Orthogonal Polynomials . 71 4.5 The Three-Term Recurrence . 72 4.6 Numerical Integration . 73 5 Eigenthings 77 5.1 Self-Adjoint Operators . 77 5.2 Diagonalizability . 78 5.3 Diagonalizability, Again . 79 5.4 Eigenvectors and Optimization . 80 5.5 The Singular Value Decomposition . 81 5.6 Least Squares . 83 5.7 Legendre Polynomials . 85 5.8 Computing Eigenvalues . 87 5.9 Jacobi: An Example . 89 6 Spectral Decomposition 91 6.1 Self-Adjoint Operators . 91 6.2 Commutative Algebras . 92 6.3 Normal Operators . 94 CONTENTS vii 7 Norms 97 7.1 Convexity . 97 7.2 Extreme Points . 99 7.3 Norms . 100 7.4 Dual Norms . 102 7.5 Matrix Norms . 103 7.6 Examples . 105 7.7 Matrix Functions . 106 7.8 Powers ................................... 108 7.9 Contractions . 110 7.10 Projections . 112 7.11 Contractions . 114 7.12 Perron . 116 8 Geometry 121 8.1 Semilinear Forms . 121 8.2 The Classification of Forms . 122 8.3 Gram Matrices . 123 8.4 Equiangular Lines . 124 8.5 Tight Frames . 125 8.6 Another Gram Matrix . 126 8.7 The Orthogonal Group . 127 8.8 Skew-Symmetric to Orthogonal . 128 8.9 Reflections . 129 9 Positive Semidefinite Matrices 133 9.1 Factorizing Positive Semidefinite Matrices . 133 9.2 Computing Cholesky . 135 9.3 Polynomial Examples . 136 9.4 Positive Semidefinite Matrices . 138 10 Tensors 141 10.1 Tensor Products . 141 10.2 Quadratic Tensors . 142 10.3 Cubic Tensors . 144 10.4 Multiplication . 146 10.5 Semifields . 147 viii CONTENTS 11 Control 149 11.1 Buffalos . 149 11.2 Burgers . 151 11.3 Controllability . 152 11.4 Observability . 155 11.5 Feedback and Controllability . 156 11.6 Canonical Forms . 158 11.7 Eigenvalues and Controllability . 159 11.8 Observers . 161 11.9 Transfer Matrices . 162 12 The Smith Normal Form 165 12.1 Domains . 165 12.2 Localization . 167 12.3 Fitting . 168 12.4 Hermite . 169 12.5 Smith Normal Form . 171 13 Polynomial and Rational Matrices 175 13.1 Series . 176 13.2 Polynomial Matrices . 177 13.3 Paraunitary Matrices . 180 13.4 Division . 182 13.5 Cayley-Hamilton . 183 13.6 Greatest Common Divisors . 185 13.7 An Identity . 187 13.8 Resolvents . 189 14 Determinants 195 14.1 Permutations . 195 14.2 The Sign of a Permutation . 197 14.3 Permutation Matrices . 198 14.4 Definition of the Determinant . ..
Recommended publications
  • On Finding Multiplicities of Characteristic Polynomial Factors Of
    On finding multiplicities of characteristic polynomial factors of black-box matrices∗. Jean-Guillaume Dumas† Cl´ement Pernet‡ B. David Saunders§ November 2, 2018 Abstract We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is rep- resented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory. Keywords: Characteristic polynomial ; black-box matrix ; finite field. 1 Introduction Computing the characteristic polynomial of an integer matrix is a classical math- ematical problem. It is closely related to the computation of the Frobenius nor- mal form which can be used to test two matrices for similarity, or computing invariant subspaces under the action of the matrix. Although the Frobenius normal form contains more information on the matrix than the characteristic arXiv:0901.4747v2 [cs.SC] 18 May 2009 polynomial, most algorithms to compute it are based on computations of char- acteristic polynomials (see for example [25, 9.7]). Several matrix representations are used in§ computational linear algebra. In the dense representation, a m n matrix is considered as the array of all the m n coefficients. The sparse representation× only considers non-zero coefficients using× different possible data structures. In the black-box representation, the matrix ∗Saunders supported by National Science Foundation Grants CCF-0515197, CCF-0830130.
    [Show full text]
  • A Weakly Stable Algorithm for General Toeplitz Systems
    A Weakly Stable Algorithm for General Toeplitz Systems∗ Adam W. Bojanczyk Richard P. Brent School of Electrical Engineering Computer Sciences Laboratory Cornell University Australian National University Ithaca, NY 14853-5401 Canberra, ACT 0200 Frank R. de Hoog Division of Mathematics and Statistics CSIRO, GPO Box 1965 Canberra, ACT 2601 Report TR-CS-93-15 6 August 1993 (revised 24 June 1994) Abstract We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that RT R is close to AT A. Thus, when the algorithm is used to solve the semi-normal equations RT Rx = AT b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min kAx − bk2. 1991 Mathematics Subject Classification. Primary 65F25; Secondary 47B35, 65F05, 65F30, 65Y05, 65Y10 Key words and phrases. Cholesky factorization, error analysis, Hankel matrix, least squares, normal equations, orthogonal factorization, QR factorization, semi-normal equa- tions, stability, Toeplitz matrix, weak stability. 1 Introduction arXiv:1005.0503v1 [math.NA] 4 May 2010 Toeplitz linear systems arise in many applications, and there are many algorithms which solve nonsingular n × n Toeplitz systems Ax = b in O(n2) arithmetic operations [2, 13, 44, 45, 49, 54, 55, 65, 67, 77, 78, 80, 81, 85, 86]. Some algorithms are restricted to symmetric systems (A = AT ) and others apply to general Toeplitz systems. Because of their recursive nature, most O(n2) algorithms assume that all leading principal submatrices of A are nonsingular, and break down if this is not the case.
    [Show full text]
  • Linear Algebra: Beware!
    LINEAR ALGEBRA: BEWARE! MATH 196, SECTION 57 (VIPUL NAIK) You might be expecting linear algebra to be a lot like your calculus classes at the University. This is probably true in terms of the course structure and format. But it’s not true at the level of subject matter. Some important differences are below. • Superficially, linear algebra is a lot easier, since it relies mostly on arithmetic rather than algebra. The computational procedures involve the systematic and correct application of processes for adding, subtracting, multiplying, and dividing numbers. But the key word here is superficially. • Even for the apparently straightforward computational exercises, it turns out that people are able to do them a lot better if they understand what’s going on. In fact, in past test questions, people have often made fewer errors when doing the problem using full-scale algebraic symbol manipulation rather than the synthetic arithmetic method. • One important difference between linear algebra and calculus is that with calculus, it’s relatively easy to understand ideas partially. One can obtain much of the basic intuition of calculus by un- derstanding graphs of functions. In fact, limit, continuity, differentaaition, and integration all have basic descriptions in terms of the graph. Note: these aren’t fully rigorous, which is why you had to take a year’s worth of calculus class to cement your understanding of the ideas. But it’s a start. With linear algebra, there is no single compelling visual tool that connects all the ideas, and conscious effort is needed even for a partial understanding. • While linear algebra lacks any single compelling visual tool, it requires either considerable visuo- spatial skill or considerable abstract symbolic and verbal skill (or a suitable linear combination thereof).
    [Show full text]
  • Parallel Systems in Symbolic and Algebraic Computation
    UCAM-CL-TR-537 Technical Report ISSN 1476-2986 Number 537 Computer Laboratory Parallel systems in symbolic and algebraic computation Mantsika Matooane June 2002 15 JJ Thomson Avenue Cambridge CB3 0FD United Kingdom phone +44 1223 763500 http://www.cl.cam.ac.uk/ c 2002 Mantsika Matooane This technical report is based on a dissertation submitted August 2001 by the author for the degree of Doctor of Philosophy to the University of Cambridge, Trinity College. Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet: http://www.cl.cam.ac.uk/TechReports/ Series editor: Markus Kuhn ISSN 1476-2986 Abstract This thesis describes techniques that exploit the distributed memory in massively parallel processors to satisfy the peak memory requirements of some very large com- puter algebra problems. Our aim is to achieve balanced memory use, which differen- tiates this work from other parallel systems whose focus is on gaining speedup. It is widely observed that failures in computer algebra systems are mostly due to mem- ory overload: for several problems in computer algebra, some of the best available algorithms suffer from intermediate expression swell where the result is of reason- able size, but the intermediate calculation encounters severe memory limitations. This observation motivates our memory-centric approach to parallelizing computer algebra algorithms. The memory balancing is based on a randomized hashing algorithm for dynamic distribution of data. Dynamic distribution means that the intermediate data is allocated storage space at the time that it is created and therefore the system can avoid overloading some processing elements.
    [Show full text]
  • Arxiv:1810.01634V2 [Cs.SC] 6 Nov 2020 Algebraic Number Fields And
    Algebraic number fields and the LLL algorithm M. J´anos Uray [email protected] ELTE – E¨otv¨os Lor´and University (Budapest) Faculty of Informatics Department of Computer Algebra Abstract In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further than them, e.g. we also analyze some R-specific operations in K like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from Zn to Kn, and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations). 1 Introduction Exact computation with algebraic numbers is an important feature that most computer algebra systems provide. They use efficient algorithms for the calculations, arXiv:1810.01634v2 [cs.SC] 6 Nov 2020 described in several papers and books, e.g. in [1, 2, 3]. However, the computational costs of these algorithms are often not obvious to calculate, because the bit complexity depends on how much the representing multi-precision integers grow during the computation.
    [Show full text]
  • Observavility Brunovsky Normal Form: Multi-Output Linear Dynamical Systems Driss Boutat, Frédéric Kratz, Jean-Pierre Barbot
    Observavility Brunovsky Normal Form: Multi-Output Linear Dynamical Systems Driss Boutat, Frédéric Kratz, Jean-Pierre Barbot To cite this version: Driss Boutat, Frédéric Kratz, Jean-Pierre Barbot. Observavility Brunovsky Normal Form: Multi- Output Linear Dynamical Systems. American Control Conference, ACC09, IEEE, Jun 2009, St. Louis, MO, United States. hal-00772281 HAL Id: hal-00772281 https://hal.inria.fr/hal-00772281 Submitted on 10 Jan 2013 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. Observavility Brunovsky Normal Form: Multi-Output Linear Dynamical Systems Driss Boutat, Fred´ eric´ Kratz and Jean-Pierre Barbot Abstract— This paper gives the sufficient and necessary II. NOTATION AND PROBLEM STATEMENT conditions to guarantee the existence of a linear change of co- ordinates to transform a multi-output linear dynamical system (modulo a nonlinear term depending on inputs and outputs) in the observability Brunovsky canonical form. Consider the following multi-output dynamical system: I. INTRODUCTION x˙ = Ax + γ(y, u) (1) y = Cx (2) For a single output dynamical linear system the observability rank condition is a necessary and sufficient condition to transform it into the Brunovsky observability where: normal form. In this last form, it is possible to use classical observer such that [8] observer and, [5] observer.
    [Show full text]
  • Efficient Solutions to Toeplitz-Structured Linear Systems for Signal Processing
    EFFICIENT SOLUTIONS TO TOEPLITZ-STRUCTURED LINEAR SYSTEMS FOR SIGNAL PROCESSING A Dissertation Presented to The Academic Faculty by Christopher Kowalczyk Turnes In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Electrical and Computer Engineering Georgia Institute of Technology May 2014 Copyright © 2014 by Christopher Kowalczyk Turnes EFFICIENT SOLUTIONS TO TOEPLITZ-STRUCTURED LINEAR SYSTEMS FOR SIGNAL PROCESSING Approved by: Dr. James McClellan, Committee Chair Dr. Jack Poulson School of Electrical and Computer School of Computational Science and Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. Justin Romberg, Advisor Dr. Chris Barnes School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. Monson Hayes Dr. Doru Balcan School of Electrical and Computer Quantitative Finance Engineering Bank of America Georgia Institute of Technology Date Approved: May 2014 Effort is one of the things that gives meaning to life. Effort means you care about something, that something is important to you and you are willing to work for it. It would be an impoverished existence if you were not willing to value things and commit yourself to working toward them. – CAROL S. DWECK, SELF-THEORIES To my wonderful, amazing, and patient parents Cynthia and Patrick, without whom none of this would be possible. Thank you for everything (but mostly for life). ACKNOWLEDGEMENTS Foremost, I wish to express my sincere gratitude to my advisor, Dr. Justin Romberg, for his support of my work and for his patience, time, and immense knowledge. Under his direction, my technical and communication skills improved dramatically.
    [Show full text]
  • Biostatistics 615/815 Lecture 13: Programming with Matrix
    Introduction Power Matrix Matrix Computation Linear System Least square Summary Introduction Power Matrix Matrix Computation Linear System Least square Summary . Annoucements . Homework #3 . .. Biostatistics 615/815 Lecture 13: Homework 3 is due today • Programming with Matrix If you’re using Visual C++ and still have problems in using boost . • .. library, you can ask for another extension . .. Hyun Min Kang . Homework #4 . .. Homework 4 is out • February 17th, 2011 Floyd-Warshall algorithm • Note that some key information was not covered in the class. • Fair/biased coint HMM . • .. Hyun Min Kang Biostatistics 615/815 - Lecture 13 February 17th, 2011 1 / 28 Hyun Min Kang Biostatistics 615/815 - Lecture 13 February 17th, 2011 2 / 28 Introduction Power Matrix Matrix Computation Linear System Least square Summary Introduction Power Matrix Matrix Computation Linear System Least square Summary . Last lecture - Conditional independence in graphical models Markov Blanket '()!+" !" '()#*!+" #" '()$*#+" '()&*#+" '()%*#+" $" %" &" If conditioned on the variables in the gray area (variables with direct • dependency), A is independent of all the other nodes. Pr(A, C, D, E B) = Pr(A B) Pr(C B) Pr(D B) Pr(E B) A (U A πA) πA • | | | | | • ⊥ − − | Hyun Min Kang Biostatistics 615/815 - Lecture 13 February 17th, 2011 3 / 28 Hyun Min Kang Biostatistics 615/815 - Lecture 13 February 17th, 2011 4 / 28 Introduction Power Matrix Matrix Computation Linear System Least square Summary Introduction Power Matrix Matrix Computation Linear System Least square Summary
    [Show full text]
  • Faster Geometric Algorithms Via Dynamic Determinant Computation
    Faster Geometric Algorithms via Dynamic Determinant Computation Vissarion FisikopoulosK Luis Pe~narandao March 1, 2018 Abstract The computation of determinants or their signs is the core procedure in many im- portant geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total computa- tion time is consumed by these computations. In this paper we study the sequences of determinants that appear in geometric algorithms. The computation of a single determinant is accelerated by using the information from the previous computations in that sequence. We propose two dynamic determinant algorithms with quadratic arithmetic com- plexity when employed in convex hull and volume computations, and with linear arith- metic complexity when used in point location problems. We implement the proposed algorithms and perform an extensive experimental analysis. On one hand, our analysis serves as a performance study of state-of-the-art determinant algorithms and imple- mentations. On the other hand, we demonstrate the supremacy of our methods over state-of-the-art implementations of determinant and geometric algorithms. Our ex- perimental results include a 20 and 78 times speed-up in volume and point location computations in dimension 6 and 11 respectively. Keywords: determinant algorithms, orientation predicate, volume computation, rank- 1 updates, experimental analysis arXiv:1206.7067v3 [cs.CG] 12 Jan 2016 1 Introduction Computing the sign of a determinant, or in other words evaluating a determinant predicate, is in the core of many important geometric algorithms. For example, convex hull algorithms use orientation predicates, and Delaunay triangulation algorithms involve in-sphere predicates.
    [Show full text]
  • A Rough Guide to Linear Algebra
    A rough guide to linear algbera Dongryul Kim Version of April 18, 2020 ii Contents Preface v 1 Sets 1 1.1 Sets and maps . .1 1.2 Products, coproducts, and sets of maps . .4 1.3 Fun with diagrams . .7 1.4 Equivalence classes . .9 2 Vector spaces 13 2.1 Fields . 13 2.2 Vector spaces . 15 2.3 Matrices . 18 2.4 Products and direct sums . 21 2.5 Subspaces and quotients . 27 2.6 Vector spaces from linear maps . 29 2.7 Bases and dimension . 34 2.8 Dual spaces . 44 2.9 Linear algebra in combinatorics . 50 3 Multilinear algebra 55 3.1 Bilinear maps and tensor products . 55 3.2 Symmetric and exterior algebras . 62 3.3 The determinant . 68 3.4 Computing the inverse matrix . 72 4 Linear algebra without division 79 4.1 Commutative rings . 79 4.2 Modules . 82 4.3 Classification of finitely generated modules over a PID . 86 4.4 Frobenius and Jordan normal form . 97 4.5 Eigenvalues and eigenvectors . 103 iii iv CONTENTS 5 Linear algebra over R and C 109 5.1 A bit of analysis . 109 5.2 Inner products . 112 5.3 Operators on Hilbert spaces . 118 5.4 The spectral theorem . 121 5.5 Positivity of operators . 125 5.6 Duality in linear programming . 130 Epilogue 131 Preface At a certain point, one adopts a mode of learning mathematics. Then at a later point, after progressing through various styles of teaching and writing, one develops one's own point of view for how mathematics is to be taught.
    [Show full text]
  • Efficiently Calculating the Determinant of a Matrix
    Efficiently Calculating the Determinant of a Matrix Felix Limanta 13515065 Program Studi Teknik Informatika Sekolah Teknik Elektro dan Informatika Institut Teknologi Bandung, Jl. Ganesha 10 Bandung 40132, Indonesia [email protected] [email protected] Abstract—There are three commonly-used algorithms to infotype is either real or integer, depending on calculate the determinant of a matrix: Laplace expansion, LU implementation, while matrix is an arbitrarily-sized decomposition, and the Bareiss algorithm. In this paper, we array of array of infotype. first discuss the underlying mathematical principles behind the algorithms. We then realize the algorithms in pseudocode Finally, we analyze the complexity and nature of the algorithms and compare them after one another. II. BASIC MATHEMATICAL CONCEPTS Keywords—asymptotic time complexity, Bareiss algorithm, A. Determinant determinant, Laplace expansion, LU decomposition. The determinant is a real number associated with every square matrix. The determinant of a square matrix A is commonly denoted as det A, det(A), or |A|. I. INTRODUCTION Singular matrices are matrices which determinant is 0. In this paper, we describe three algorithms to find the Unimodular matrices are matrices which determinant is 1. determinant of a square matrix: Laplace expansion (also The determinant of a 1×1 matrix is the element itself. known as determinant expansion by minors), LU 퐴 = [푎] ⇒ det(퐴) = 푎 decomposition, and the Bareiss algorithm. The determinant of a 2×2 matrix is defined as the product The paper is organized as follows. Section II reviews the of the elements on the main diagonal minus the product of basic mathematical concepts for this paper, which includes the elements off the main diagonal.
    [Show full text]
  • Parallel Architecture for the Solution of Linear Equations Systems Based on Division Free Gaussian Elimination Method Implemented in FPGA
    WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS R. Martinez, D. Torres, M. Madrigal, S. Maximov Parallel Architecture for the Solution of Linear Equations Systems Based on Division Free Gaussian Elimination Method Implemented in FPGA R. MARTINEZ, D. TORRES, M. MADRIGAL, S. MAXIMOV Program of Graduated and Investigation in Electrical Engineering as the Morelia Institute Technology Av. Tecnológico 1500, Col. Lomas de Santiaguito, Morelia, Michoacán MEXICO [email protected] http://elec.itmorelia.edu.mx Abstract: - This paper presents a parallel architecture for the solution of linear equations systems based on the Division Free Gaussian Elimination Method. This architecture was implemented in a Field Programmable Gate Array (FPGA). The division-free Gaussian elimination method was integrated in identical processors in a FPGA Spartan 3 of Xilinx. A top-down design was used. The proposed architecture can handle IEEE 754 single and double precision floating-point data and the architecture was implemented in 240 processors. Also, an algorithmic complexity of O(n2) was obtained using a n2 processors scheme that perform the solution of the linear equations. Moreover, the parallel division-free Gaussian elimination method, the architecture´s data distribution and the internal processor-element (PE) architecture are presented. Finally, this paper presents the obtained simulation results and synthesis of the modules designed in very high-speed integrated circuit hardware description language (VHDL) using 40 and 100 Mhz frequencies. Key-Words: - Field Programmable Gate Array (FPGA), Parallel Processing, Parallel Architectures, linear systems equations, Division Free Gaussian elimination Method. 1 Introduction element, mathematical algorithms and power Linear systems are commonly found in many electrical systems [9].
    [Show full text]