Polynomial And Rational Matrices
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Structured Matrices And Polynomials
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Author : Victor Pan
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-06-26
Structured Matrices And Polynomials written by Victor Pan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-06-26 with Computers categories.
This user-friendly, engaging textbook makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. The book goes beyond research frontiers and, apart from very recent research articles, includes previously unpublished results.
Matrices And Matroids For Systems Analysis
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Author : Kazuo Murota
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-10-27
Matrices And Matroids For Systems Analysis written by Kazuo Murota and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10-27 with Mathematics categories.
A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the 1990's. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. From the reviews: "...The book has been prepared very carefully, contains a lot of interesting results and is highly recommended for graduate and postgraduate students." András Recski, Mathematical Reviews Clippings 2000m:93006
Counting Polynomial Matrices Over Finite Fields
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Author : Julia Lieb
language : en
Publisher: BoD – Books on Demand
Release Date : 2017-09-15
Counting Polynomial Matrices Over Finite Fields written by Julia Lieb and has been published by BoD – Books on Demand this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-15 with Mathematics categories.
This book is dealing with three mathematical areas, namely polynomial matrices over finite fields, linear systems and coding theory. Primeness properties of polynomial matrices provide criteria for the reachability and observability of interconnected linear systems. Since time-discrete linear systems over finite fields and convolutional codes are basically the same objects, these results could be transferred to criteria for non-catastrophicity of convolutional codes. In particular, formulas for the number of pairwise coprime polynomials and for the number of mutually left coprime polynomial matrices are calculated. This leads to the probability that a parallel connected linear system is reachable and that a parallel connected convolutional code is non-catastrophic. Moreover, other networks of linear systems and convolutional codes are considered.
Polynomial And Rational Matrices
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Author : Tadeusz Kaczorek
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-01-19
Polynomial And Rational Matrices written by Tadeusz Kaczorek and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-19 with Technology & Engineering categories.
This book reviews new results in the application of polynomial and rational matrices to continuous- and discrete-time systems. It provides the reader with rigorous and in-depth mathematical analysis of the uses of polynomial and rational matrices in the study of dynamical systems. It also throws new light on the problems of positive realization, minimum-energy control, reachability, and asymptotic and robust stability.
Structured Matrices And Polynomials
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Author : Victor Y. Pan
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Structured Matrices And Polynomials written by Victor Y. Pan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Structured matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms. Included are specific applications to other fields, in particular, superfast solutions to: various fundamental problems of computer algebra; the tangential Nevanlinna--Pick and matrix Nehari problems The primary intended readership for this work includes researchers, algorithm designers, and advanced graduate students in the fields of computations with structured matrices, computer algebra, and numerical rational interpolation. The book goes beyond research frontiers and, apart from very recent research articles, includes yet unpublished results. To serve a wider audience, the presentation unfolds systematically and is written in a user-friendly engaging style. Only some preliminary knowledge of the fundamentals of linear algebra is required. This makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. Examples, tables, figures, exercises, extensive bibliography, and index lend this text to classroom use or self-study.
Computation Of Generalized Matrix Inverses And Applications
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Author : Ivan Stanimirović
language : en
Publisher: CRC Press
Release Date : 2017-12-14
Computation Of Generalized Matrix Inverses And Applications written by Ivan Stanimirović and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-12-14 with Mathematics categories.
This volume offers a gradual exposition to matrix theory as a subject of linear algebra. It presents both the theoretical results in generalized matrix inverses and the applications. The book is as self-contained as possible, assuming no prior knowledge of matrix theory and linear algebra. The book first addresses the basic definitions and concepts of an arbitrary generalized matrix inverse with special reference to the calculation of {i,j,...,k} inverse and the Moore–Penrose inverse. Then, the results of LDL* decomposition of the full rank polynomial matrix are introduced, along with numerical examples. Methods for calculating the Moore–Penrose’s inverse of rational matrix are presented, which are based on LDL* and QDR decompositions of the matrix. A method for calculating the A(2)T;S inverse using LDL* decomposition using methods is derived as well as the symbolic calculation of A(2)T;S inverses using QDR factorization. The text then offers several ways on how the introduced theoretical concepts can be applied in restoring blurred images and linear regression methods, along with the well-known application in linear systems. The book also explains how the computation of generalized inverses of matrices with constant values is performed. It covers several methods, such as methods based on full-rank factorization, Leverrier–Faddeev method, method of Zhukovski, and variations of the partitioning method.
Inverse Problems For Polynomial And Rational Matrices
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Author : Richard Allen Hollister
language : en
Publisher:
Release Date : 2020
Inverse Problems For Polynomial And Rational Matrices written by Richard Allen Hollister and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Eigenvalues categories.
Inverse problems have long been studied in mathematics not only because there are many applications in science and engineering, but also because they yield new insight into the beauty of mathematics. Central to the subject of linear algebra is the eigenvalue problem: given a matrix, and its eigenvalues (numerical invariants). Eigenvalue problems play a key role in almost every field of scientific endeavor from calculating the vibrational modes of a molecule to modeling the spread of an infectious disease, and so have been studied extensively since the time of Euler in the 18th century. If a typical matrix eigenvalue problem asks for the eigenvalues of a given matrix, an inverse eigenvalue problem asks for a matrix whose eigenvalues are a given list of numbers. For matrices over an algebraically closed field, the inverse eigenvalue problem is completely and transparently solved by the Jordan canonical form. If the field is not algebraically closed, there are similar, albeit more involved, solutions, a prime example of which is the real Jordan form when the field is the real numbers. Eigenvalue and inverse eigenvalue problems go beyond just matrices with fixed scalar entries. They have been studied for matrix pencils, which are matrices whose entries are degree-one polynomials with coefficients from a field. A polynomial matrix is a matrix whose entries are polynomials with coefficients from a field. The story of eigenvalues for polynomial matrices (of which matrix pencils are a special case) is more complicated because of the possibility of an infinite eigenvalue. In addition, for singular polynomial matrices, there are invariants that characterize the left and right null spaces called minimal indices. The collection of all this data (finite and infinite eigenvalues together with minimal indices) is known as the structural data of the polynomial matrix. In this dissertation, the inverse structural data problem for polynomial matrices is considered and solved. We begin with the history of this inverse problem, including known results and applications from the literature. Then a new solution is given that is sparse and transparently reveals the structural data in much the same way that the Jordan canonical form transparently reveals the structural data of a scalar matrix. The dissertation concludes by discussing the inverse problem for rational matrices (matrices whose entries are rational functions over a field) and presenting a solution adapted from the solution for the polynomial matrix inverse problem.
Orthogonal Rational Functions
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Author : Adhemar Bultheel
language : en
Publisher: Cambridge University Press
Release Date : 1999-02-13
Orthogonal Rational Functions written by Adhemar Bultheel and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-02-13 with Mathematics categories.
This book generalises the classical theory of orthogonal polynomials on the complex unit circle, or on the real line to orthogonal rational functions whose poles are among a prescribed set of complex numbers. The first part treats the case where these poles are all outside the unit disk or in the lower half plane. Classical topics such as recurrence relations, numerical quadrature, interpolation properties, Favard theorems, convergence, asymptotics, and moment problems are generalised and treated in detail. The same topics are discussed for the different situation where the poles are located on the unit circle or on the extended real line. In the last chapter, several applications are mentioned including linear prediction, Pisarenko modelling, lossless inverse scattering, and network synthesis. This theory has many applications in theoretical real and complex analysis, approximation theory, numerical analysis, system theory, and in electrical engineering.
Matrix Polynomials
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Author : I. Gohberg
language : en
Publisher: SIAM
Release Date : 2009-07-23
Matrix Polynomials written by I. Gohberg and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-07-23 with Mathematics categories.
This book is the definitive treatment of the theory of polynomials in a complex variable with matrix coefficients. Basic matrix theory can be viewed as the study of the special case of polynomials of first degree; the theory developed in Matrix Polynomials is a natural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations, systems theory, the Wiener-Hopf technique, mechanics and vibrations, and numerical analysis. Although there have been significant advances in some quarters, this work remains the only systematic development of the theory of matrix polynomials. The book is appropriate for students, instructors, and researchers in linear algebra, operator theory, differential equations, systems theory, and numerical analysis. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex analysis.
Matrix Analysis
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Author : Roger A. Horn
language : en
Publisher: Cambridge University Press
Release Date : 1990-02-23
Matrix Analysis written by Roger A. Horn and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990-02-23 with Mathematics categories.
In this book the authors present classical and recent results for matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematics, and the necessary material has only occurred sporadically in the literature and university curricula. As the interest in applied mathematics has grown, the need for a text and a reference work offering a broad selection of topics has become apparent, and this book aims to meet that need. This book will be welcomed as an undergraduate or graduate textbook for students studying matrix analysis. The authors assume a background in elementary linear algebra and knowledge of rudimentary analytical concepts. They begin with a review and discussion of eigenvalues and eigenvectors. The following chapters each treat a major topic in depth. This volume should be useful not only as a text, but also as a self-contained reference work to a variety of audiences in other scientific fields.