Problems and Theorems in Linear Algebra
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Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix
Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix André Klein University of Amsterdam Guy Mélard ECARES, Université libre de Bruxelles April 2020 ECARES working paper 2020-11 ECARES ULB - CP 114/04 50, F.D. Roosevelt Ave., B-1050 Brussels BELGIUM www.ecares.org Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix Andr´eKlein∗ Guy M´elard† Abstract In this paper the invertibility condition of the asymptotic Fisher information matrix of a controlled vector autoregressive moving average stationary process, VARMAX, is dis- played in a theorem. It is shown that the Fisher information matrix of a VARMAX process becomes invertible if the VARMAX matrix polynomials have no common eigenvalue. Con- trarily to what was mentioned previously in a VARMA framework, the reciprocal property is untrue. We make use of tensor Sylvester matrices since checking equality of the eigen- values of matrix polynomials is most easily done in that way. A tensor Sylvester matrix is a block Sylvester matrix with blocks obtained by Kronecker products of the polynomial coefficients by an identity matrix, on the left for one polynomial and on the right for the other one. The results are illustrated by numerical computations. MSC Classification: 15A23, 15A24, 60G10, 62B10. Keywords: Tensor Sylvester matrix; Matrix polynomial; Common eigenvalues; Fisher in- formation matrix; Stationary VARMAX process. Declaration of interest: none ∗University of Amsterdam, The Netherlands, Rothschild Blv. 123 Apt.7, 6527123 Tel-Aviv, Israel (e-mail: [email protected]). †Universit´elibre de Bruxelles, Solvay Brussels School of Economics and Management, ECARES, avenue Franklin Roosevelt 50 CP 114/04, B-1050 Brussels, Belgium (e-mail: [email protected]). -
Beam Modes of Lasers with Misaligned Complex Optical Elements
Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 1995 Beam Modes of Lasers with Misaligned Complex Optical Elements Anthony A. Tovar Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Tovar, Anthony A., "Beam Modes of Lasers with Misaligned Complex Optical Elements" (1995). Dissertations and Theses. Paper 1363. https://doi.org/10.15760/etd.1362 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. DISSERTATION APPROVAL The abstract and dissertation of Anthony Alan Tovar for the Doctor of Philosophy in Electrical and Computer Engineering were presented April 21, 1995 and accepted by the dissertation committee and the doctoral program. Carl G. Bachhuber Vincent C. Williams, Representative of the Office of Graduate Studies DOCTORAL PROGRAM APPROVAL: Rolf Schaumann, Chair, Department of Electrical Engineering ************************************************************************ ACCEPTED FOR PORTLAND STATE UNIVERSITY BY THE LIBRARY ABSTRACT An abstract of the dissertation of Anthony Alan Tovar for the Doctor of Philosophy in Electrical and Computer Engineering presented April 21, 1995. Title: Beam Modes of Lasers with Misaligned Complex Optical Elements A recurring theme in my research is that mathematical matrix methods may be used in a wide variety of physics and engineering applications. Transfer matrix tech niques are conceptually and mathematically simple, and they encourage a systems approach. -
PSEUDO SCHUR COMPLEMENTS, PSEUDO PRINCIPAL PIVOT TRANSFORMS and THEIR INHERITANCE PROPERTIES∗ 1. Introduction. Let M ∈ R M×
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 455-477, August 2015 ELA PSEUDO SCHUR COMPLEMENTS, PSEUDO PRINCIPAL PIVOT TRANSFORMS AND THEIR INHERITANCE PROPERTIES∗ K. BISHT† , G. RAVINDRAN‡, AND K.C. SIVAKUMAR§ Abstract. Extensions of the Schur complement and the principal pivot transform, where the usual inverses are replaced by the Moore-Penrose inverse, are revisited. These are called the pseudo Schur complement and the pseudo principal pivot transform, respectively. First, a generalization of the characterization of a block matrix to be an M-matrix is extended to the nonnegativity of the Moore-Penrose inverse. A comprehensive treatment of the fundamental properties of the extended notion of the principal pivot transform is presented. Inheritance properties with respect to certain matrix classes are derived, thereby generalizing some of the existing results. Finally, a thorough discussion on the preservation of left eigenspaces by the pseudo principal pivot transformation is presented. Key words. Principal pivot transform, Schur complement, Nonnegative Moore-Penrose inverse, P†-Matrix, R†-Matrix, Left eigenspace, Inheritance properties. AMS subject classifications. 15A09, 15A18, 15B48. 1. Introduction. Let M ∈ Rm×n be a block matrix partitioned as A B , C D where A ∈ Rk×k is nonsingular. Then the classical Schur complement of A in M denoted by M/A is given by D − CA−1B ∈ R(m−k)×(n−k). This notion has proved to be a fundamental idea in many applications like numerical analysis, statistics and operator inequalities, to name a few. We refer the reader to [27] for a comprehen- sive account of the Schur complement. -
Generalized Sylvester Theorems for Periodic Applications in Matrix Optics
Portland State University PDXScholar Electrical and Computer Engineering Faculty Publications and Presentations Electrical and Computer Engineering 3-1-1995 Generalized Sylvester theorems for periodic applications in matrix optics Lee W. Casperson Portland State University Anthony A. Tovar Follow this and additional works at: https://pdxscholar.library.pdx.edu/ece_fac Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Citation Details Anthony A. Tovar and W. Casperson, "Generalized Sylvester theorems for periodic applications in matrix optics," J. Opt. Soc. Am. A 12, 578-590 (1995). This Article is brought to you for free and open access. It has been accepted for inclusion in Electrical and Computer Engineering Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. 578 J. Opt. Soc. Am. AIVol. 12, No. 3/March 1995 A. A. Tovar and L. W. Casper Son Generalized Sylvester theorems for periodic applications in matrix optics Anthony A. Toyar and Lee W. Casperson Department of Electrical Engineering,pirtland State University, Portland, Oregon 97207-0751 i i' Received June 9, 1994; revised manuscripjfeceived September 19, 1994; accepted September 20, 1994 Sylvester's theorem is often applied to problems involving light propagation through periodic optical systems represented by unimodular 2 X 2 transfer matrices. We extend this theorem to apply to broader classes of optics-related matrices. These matrices may be 2 X 2 or take on an important augmented 3 X 3 form. The results, which are summarized in tabular form, are useful for the analysis and the synthesis of a variety of optical systems, such as those that contain periodic distributed-feedback lasers, lossy birefringent filters, periodic pulse compressors, and misaligned lenses and mirrors. -
A Note on the Inversion of Sylvester Matrices in Control Systems
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2011, Article ID 609863, 10 pages doi:10.1155/2011/609863 Research Article A Note on the Inversion of Sylvester Matrices in Control Systems Hongkui Li and Ranran Li College of Science, Shandong University of Technology, Shandong 255049, China Correspondence should be addressed to Hongkui Li, [email protected] Received 21 November 2010; Revised 28 February 2011; Accepted 31 March 2011 Academic Editor: Gradimir V. Milovanovic´ Copyright q 2011 H. Li and R. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a sufficient condition the solvability of two standard equations of Sylvester matrix by using the displacement structure of the Sylvester matrix, and, according to the sufficient condition, we derive a new fast algorithm for the inversion of a Sylvester matrix, which can be denoted as a sum of products of two triangular Toeplitz matrices. The stability of the inversion formula for a Sylvester matrix is also considered. The Sylvester matrix is numerically forward stable if it is nonsingular and well conditioned. 1. Introduction Let Rx be the space of polynomials over the real numbers. Given univariate polynomials ∈ f x ,g x R x , a1 / 0, where n n−1 ··· m m−1 ··· f x a1x a2x an1,a1 / 0,gx b1x b2x bm1,b1 / 0. 1.1 Let S denote the Sylvester matrix of f x and g x : ⎫ ⎛ ⎞ ⎪ ⎪ a a ··· ··· a ⎪ ⎜ 1 2 n 1 ⎟ ⎬ ⎜ ··· ··· ⎟ ⎜ a1 a2 an1 ⎟ m row ⎜ ⎟ ⎪ ⎜ . -
(Integer Or Rational Number) Arithmetic. -Computing T
Electronic Transactions on Numerical Analysis. ETNA Volume 18, pp. 188-197, 2004. Kent State University Copyright 2004, Kent State University. [email protected] ISSN 1068-9613. A NEW SOURCE OF STRUCTURED SINGULAR VALUE DECOMPOSITION PROBLEMS ¡ ANA MARCO AND JOSE-J´ AVIER MARTINEZ´ ¢ Abstract. The computation of the Singular Value Decomposition (SVD) of structured matrices has become an important line of research in numerical linear algebra. In this work the problem of inversion in the context of the computation of curve intersections is considered. Although this problem has usually been dealt with in the field of exact rational computations and in that case it can be solved by using Gaussian elimination, when one has to work in finite precision arithmetic the problem leads to the computation of the SVD of a Sylvester matrix, a different type of structured matrix widely used in computer algebra. In addition only a small part of the SVD is needed, which shows the interest of having special algorithms for this situation. Key words. curves, intersection, singular value decomposition, structured matrices. AMS subject classifications. 14Q05, 65D17, 65F15. 1. Introduction and basic results. The singular value decomposition (SVD) is one of the most valuable tools in numerical linear algebra. Nevertheless, in spite of its advantageous features it has not usually been applied in solving curve intersection problems. Our aim in this paper is to consider the problem of inversion in the context of the computation of the points of intersection of two plane curves, and to show how this problem can be solved by computing the SVD of a Sylvester matrix, a type of structured matrix widely used in computer algebra. -
Evaluation of Spectrum of 2-Periodic Tridiagonal-Sylvester Matrix
Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix Emrah KILIÇ AND Talha ARIKAN Abstract. The Sylvester matrix was …rstly de…ned by J.J. Sylvester. Some authors have studied the relationships between certain or- thogonal polynomials and determinant of Sylvester matrix. Chu studied a generalization of the Sylvester matrix. In this paper, we introduce its 2-period generalization. Then we compute its spec- trum by left eigenvectors with a similarity trick. 1. Introduction There has been increasing interest in tridiagonal matrices in many di¤erent theoretical …elds, especially in applicative …elds such as numer- ical analysis, orthogonal polynomials, engineering, telecommunication system analysis, system identi…cation, signal processing (e.g., speech decoding, deconvolution), special functions, partial di¤erential equa- tions and naturally linear algebra (see [2, 4, 5, 6, 15]). Some authors consider a general tridiagonal matrix of …nite order and then describe its LU factorizations, determine the determinant and inverse of a tridi- agonal matrix under certain conditions (see [3, 7, 10, 11]). The Sylvester type tridiagonal matrix Mn(x) of order (n + 1) is de- …ned as x 1 0 0 0 0 n x 2 0 0 0 2 . 3 0 n 1 x 3 .. 0 0 6 . 7 Mn(x) = 6 . .. .. .. .. 7 6 . 7 6 . 7 6 0 0 0 .. .. n 1 0 7 6 7 6 0 0 0 0 x n 7 6 7 6 0 0 0 0 1 x 7 6 7 4 5 1991 Mathematics Subject Classi…cation. 15A36, 15A18, 15A15. Key words and phrases. Sylvester Matrix, spectrum, determinant. Corresponding author. -
Notes on the Schur Complement
University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 12-10-2010 Notes on the Schur Complement Jean H. Gallier University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Computer Sciences Commons Recommended Citation Jean H. Gallier, "Notes on the Schur Complement", . December 2010. Gallier, J., Notes on the Schur Complement This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/601 For more information, please contact [email protected]. Notes on the Schur Complement Disciplines Computer Sciences Comments Gallier, J., Notes on the Schur Complement This working paper is available at ScholarlyCommons: https://repository.upenn.edu/cis_papers/601 The Schur Complement and Symmetric Positive Semidefinite (and Definite) Matrices Jean Gallier December 10, 2010 1 Schur Complements In this note, we provide some details and proofs of some results from Appendix A.5 (especially Section A.5.5) of Convex Optimization by Boyd and Vandenberghe [1]. Let M be an n × n matrix written a as 2 × 2 block matrix AB M = ; CD where A is a p × p matrix and D is a q × q matrix, with n = p + q (so, B is a p × q matrix and C is a q × p matrix). We can try to solve the linear system AB x c = ; CD y d that is Ax + By = c Cx + Dy = d; by mimicking Gaussian elimination, that is, assuming that D is invertible, we first solve for y getting y = D−1(d − Cx) and after substituting this expression for y in the first equation, we get Ax + B(D−1(d − Cx)) = c; that is, (A − BD−1C)x = c − BD−1d: 1 If the matrix A − BD−1C is invertible, then we obtain the solution to our system x = (A − BD−1C)−1(c − BD−1d) y = D−1(d − C(A − BD−1C)−1(c − BD−1d)): The matrix, A − BD−1C, is called the Schur Complement of D in M. -
The Schur Complement and H-Matrix Theory
UNIVERSITY OF NOVI SAD FACULTY OF TECHNICAL SCIENCES Maja Nedović THE SCHUR COMPLEMENT AND H-MATRIX THEORY DOCTORAL DISSERTATION Novi Sad, 2016 УНИВЕРЗИТЕТ У НОВОМ САДУ ФАКУЛТЕТ ТЕХНИЧКИХ НАУКА 21000 НОВИ САД, Трг Доситеја Обрадовића 6 КЉУЧНА ДОКУМЕНТАЦИЈСКА ИНФОРМАЦИЈА Accession number, ANO: Identification number, INO: Document type, DT: Monographic publication Type of record, TR: Textual printed material Contents code, CC: PhD thesis Author, AU: Maja Nedović Mentor, MN: Professor Ljiljana Cvetković, PhD Title, TI: The Schur complement and H-matrix theory Language of text, LT: English Language of abstract, LA: Serbian, English Country of publication, CP: Republic of Serbia Locality of publication, LP: Province of Vojvodina Publication year, PY: 2016 Publisher, PB: Author’s reprint Publication place, PP: Novi Sad, Faculty of Technical Sciences, Trg Dositeja Obradovića 6 Physical description, PD: 6/161/85/1/0/12/0 (chapters/pages/ref./tables/pictures/graphs/appendixes) Scientific field, SF: Applied Mathematics Scientific discipline, SD: Numerical Mathematics Subject/Key words, S/KW: H-matrices, Schur complement, Eigenvalue localization, Maximum norm bounds for the inverse matrix, Closure of matrix classes UC Holding data, HD: Library of the Faculty of Technical Sciences, Trg Dositeja Obradovića 6, Novi Sad Note, N: Abstract, AB: This thesis studies subclasses of the class of H-matrices and their applications, with emphasis on the investigation of the Schur complement properties. The contributions of the thesis are new nonsingularity results, bounds for the maximum norm of the inverse matrix, closure properties of some matrix classes under taking Schur complements, as well as results on localization and separation of the eigenvalues of the Schur complement based on the entries of the original matrix. -
Uniqueness of Solution of a Generalized ⋆-Sylvester Matrix
Uniqueness of solution of a generalized ?-Sylvester matrix equationI Fernando De Ter´ana,∗, Bruno Iannazzob aDepartamento de Matem´aticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Legan´es,Spain. [email protected]. bDipartimento di Matematica e Informatica, Universit`adi Perugia, Via Vanvitelli 1, 06123 Perugia, Italy. [email protected] Abstract We present necessary and sufficient conditions for the existence of a unique solution of the generalized ?-Sylvester matrix equation AXB + CX?D = E, where A; B; C; D; E are square matrices of the same size with real or complex entries, and where ? stands for either the transpose or the conjugate transpose. This generalizes several previous uniqueness results for specific equations like the ?-Sylvester or the ?-Stein equations. Keywords: Linear matrix equation, matrix pencil, Sylvester equation, ?-Sylvester equation, ?-Stein equation, T -Sylvester equation, eigenvalues. AMS classification: 15A22, 15A24, 65F15 1. Introduction Given A; B; C; D; E 2 Fn×n, with F being C or R, we consider the equa- tion AXB + CX?D = E (1) where X 2 Fn×n is an unknown matrix, and where, for a given matrix M 2 Fn×n, M ? stands for either the transpose M T or the conjugate transpose M ∗. IThis work was partially supported by the Ministerio de Econom´ıay Competitividad of Spain through grant MTM-2012-32542 (F. De Ter´an)and by the INdAM through a GNCS Project 2015 (B. Iannazzo). It was partially developed while F. De Ter´anwas visiting the Universit`adi Perugia, funded by INdAM. ∗corresponding author. Preprint submitted to Linear Algebra and its Applications October 20, 2015 In the last few years, this equation has been considered by several authors in the context of linear Sylvester-like equations arising in applications (see, for instance, [3, 14, 17, 18]). -
Matrix Algebraic Properties of the Fisher Information Matrix of Stationary Processes
Entropy 2014, 16, 2023-2055; doi:10.3390/e16042023 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Matrix Algebraic Properties of the Fisher Information Matrix of Stationary Processes Andre´ Klein Rothschild Blv. 123 Apt.7, 65271 Tel Aviv, Israel; E-Mail: [email protected] or [email protected]; Tel.: 972.5.25594723 Received: 12 February 2014; in revised form: 11 March 2014 / Accepted: 24 March 2014 / Published: 8 April 2014 Abstract: In this survey paper, a summary of results which are to be found in a series of papers, is presented. The subject of interest is focused on matrix algebraic properties of the Fisher information matrix (FIM) of stationary processes. The FIM is an ingredient of the Cramer-Rao´ inequality, and belongs to the basics of asymptotic estimation theory in mathematical statistics. The FIM is interconnected with the Sylvester, Bezout and tensor Sylvester matrices. Through these interconnections it is shown that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property. A statistical distance measure involving entries of the FIM is presented. In quantum information, a different statistical distance measure is set forth. It is related to the Fisher information but where the information about one parameter in a particular measurement procedure is considered. The FIM of scalar stationary processes is also interconnected to the solutions of appropriate Stein equations, conditions for the FIM to verify certain Stein equations are formulated. The presence of Vandermonde matrices is also emphasized. Keywords: Bezout matrix; Sylvester matrix; tensor Sylvester matrix; Stein equation; Vandermonde matrix; stationary process; matrix resultant; Fisher information matrix MSC Classification: 15A23, 15A24, 15B99, 60G10, 62B10, 62M20. -
Polynomial Bezoutian Matrix with Respect to a General Basisୋ Z.H
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 331 (2001) 165–179 www.elsevier.com/locate/laa Polynomial Bezoutian matrix with respect to a general basisୋ Z.H. Yang Department of Mathematics, East Campus, China Agricultural University, P.O. Box 71, 100083 Beijing, People’s Republic of China Received 7 September 2000; accepted 25 January 2001 Submitted by R.A. Brualdi Abstract We introduce a class of so-called polynomial Bezoutian matrices. The operator represen- tation relative to a pair of dual bases and the generalized Barnett factorization formula for this kind of matrix are derived. An intertwining relation and generalized Bezoutian reduction via polynomial Vandermonde matrix are presented. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A57 Keywords: Polynomial Bezoutian; Vandermonde matrix; Barnett-type formula 1. Introduction Given a pair of polynomials p(z) and q(z) with degree p(z) = n and degree q(z) n, the (classical) Bezoutian of p(z) and q(z) is the bilinear form given by − − p(x)q(y) − p(y)q(x) n 1 n 1 π(x,y) = = b xiyj (1.1) x − y ij i=0 j=0 × = and Bezoutian matrix of p and q is defined by the n n symmetric matrix B(p, q) n−1 bij i,j=0. ୋ Partially supported by National Science Foundation of China. 0024-3795/01/$ - see front matter 2001 Elsevier Science Inc. All rights reserved. PII:S0024-3795(01)00282-8 166 Z.H.