DOCSLIB.ORG

Mmono134-Endmatter.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mmono134-Endmatter.Pdf Selected Titles in This Series 151 S . Yu . Slavyanov , Asymptotic solutions o f the one-dimensional Schrodinge r equation , 199 6 150 B . Ya. Levin , Lectures on entire functions, 199 6 149 Takash i Sakai, Riemannian geometry , 199 6 148 Vladimi r I. Piterbarg, Asymptotic methods i n the theory o f Gaussian processe s and fields, 1996 147 S . G . Gindiki n an d L. R. Volevich , Mixed problem fo r partia l differentia l equation s with quasihomogeneous principa l part, 199 6 146 L . Ya. Adrianova , Introduction t o linear system s of differential equations , 199 5 145 A . N. Andrianov and V. G. Zhuravlev , Modular form s an d Heck e operators, 199 5 144 O . V. Troshkin, Nontraditional method s in mathematical hydrodynamics , 199 5 143 V . A. Malyshev an d R. A. Minlos, Linear infinite-particl e operators , 199 5 142 N . V . Krylov, Introduction t o the theory o f diffusio n processes , 199 5 141 A . A. Davydov, Qualitative theor y o f control systems , 199 4 140 Aizi k I. Volpert, Vitaly A. Volpert, an d Vladimir A. Volpert, Traveling wav e solutions o f parabolic systems, 199 4 139 I . V . Skrypnik, Methods for analysi s o f nonlinear ellipti c boundary valu e problems, 199 4 138 Yu . P. Razmyslov, Identities o f algebras and their representations, 199 4 137 F . I. Karpelevich and A. Ya. Kreinin , Heavy traffic limit s for multiphas e queues, 199 4 136 Masayosh i Miyanishi, Algebraic geometry, 199 4 135 Masar u Takeuchi, Modern spherica l functions, 199 4 134 V . V. Prasolov, Problems and theorem s in linear algebra, 199 4 133 P . I. Naumkin an d I. A. Shishmarev, Nonlinear nonloca l equations in the theory o f waves, 199 4 132 Hajim e Urakawa, Calculus of variations an d harmonic maps, 199 3 131 V . V. Sharko, Functions on manifolds: Algebrai c and topological aspects, 199 3 130 V . V. Vershinin, Cobordisms and spectral sequences , 199 3 129 Mitsu o Morimoto, An introduction t o Sato' s hyperfunctions, 199 3 128 V . P. Orevkov, Complexity o f proofs an d their transformations i n axiomatic theories, 199 3 127 F . L. Zak, Tangents and secants o f algebraic varieties, 199 3 126 M . L. Agranovskii, Invariant functio n space s o n homogeneous manifold s o f Lie groups and applications, 199 3 125 Masayosh i Nagata, Theory o f commutative fields, 1993 124 Masahis a Adachi, Embeddings an d immersions, 199 3 123 M . A. Akivis and B. A. Rosenfeld, Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou, Theory o f entire and meromorphic functions : Deficien t an d asymptoti c value s and singula r directions, 199 3 121 LB . Fesenk o an d S. V . Vostokov, Local fields and thei r extensions: A constructive approach, 199 3 120 Takeyuk i Hid a an d Masuyuki Hitsuda, Gaussian processes , 199 3 119 M . V . Karasev an d V. P. Maslov, Nonlinear Poisso n brackets. Geometry an d quantization, 199 3 118 Kenkich i Iwasawa, Algebraic functions, 199 3 117 Bori s Zilber, Uncountably categorica l theories , 199 3 116 G . M. Fel'dman, Arithmetic of probability distributions , and characterization problem s on abelian groups, 199 3 115 Nikola i V . Ivanov, Subgroups o f Teichmuller modular groups , 199 2 114 Seiz o ltd, Diffusion equations , 199 2 113 Michai l Zhitomirskil , Typical singularities o f differential 1-form s and Pfaffia n equations , 199 2 112 S . A. Lomov, Introduction t o the general theory o f singular perturbations, 199 2 111 Simo n Gindikin, Tube domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introduction t o comple x analysis Part II. Functions o f severa l variables, 199 2 (Continued in the back of this publication) Problems and Theorems in Linear Algebra This page intentionally left blank V.V. Prasoiov Problems and Theorems in Linear Algebra 10.1090/mmono/134 BHKTOP BacHJibeBir a IIpacojiO B 3AHAMH H TEOPEML I JIHHEHHOH AJirEBPb l Translated b y D . A . Leite s fro m a n origina l Russia n manuscrip t Translation edite d b y Simeo n Ivano v 2000 Mathematics Subject Classification. Primar y 15-01 . ABSTRACT. Thi s boo k contain s the basic s o f linear algebr a with a n emphasi s o n nonstandard an d neat proof s o f known theorems. Man y o f the theorems o f linear algebra obtained mainly during th e past thirt y year s ar e usuall y ignore d i n textbooks bu t ar e quit e accessibl e fo r student s majorin g or minorin g i n mathematics . Thes e theorem s ar e give n wit h complet e proofs . Ther e ar e abou t 230 problems wit h solutions . Library o f Congres s Cataloging-in-Publicatio n Dat a Prasolov, V . V . (Vikto r Vasil'evich ) [Zadachi i teoremy linemo i algebry . English ] Problems an d theorem s i n linea r algebra/V . V . Prasolov ; [translate d b y D . A . Leite s fro m a n original Russia n manuscript ; translatio n edite d b y Simeo n Ivanov ] p. cm . — (Translation s o f mathematical monographs , ISS N 0065-9282 ; v. 134 ) Includes bibliographica l references . ISBN 0-8218-0236- 4 1. Algebras , Linear . I . Ivanov , Simeon . II . Title . III . Series . QA184.P7313 199 4 94-1333 2 512/.5—dc20 CI P 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 mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed to the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-229 4 USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 199 4 b y the America n Mathematica l Society . Al l rights reserved . Reprinted wit h correction s 1996 . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . & Printe d o n recycle d paper . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 1 3 1 2 1 1 1 0 0 9 0 8 CONTENTS Preface x v Main notations and conventions xvi i Chapter I. Determinant s 1 Historical remarks: Leibni z and Sek i Kova. Cramer , L'Hospital, Cauch y and Jacobi 1. Basic properties of determinants 1 The Vandermonde determinant an d its application. Th e Cauchy determinant. Continue d frac - tions and the determinant of a tridiagonal matrix. Certai n other determinants. Problems 2. Minors and cofactors 9 Binet-Cauchy's formula. Laplace' s theorem. Jacobi' s theorem on minors of the adjoint matrix. The generalize d Sylvester' s identity . Chebotarev' s theore m o n th e matri x ||£ u||f~ , wher e e — exp(2ni/p). Problems 3. The Schur complement 1 6 GivenA = ( n 1 2 ),thematrix(^4|^ii ) = A22—A21A7, 1 A\2 is called the Schur complement \A2\ A22J (of A\\ in A). 3.1. de t A = de t A u de t (A\A n). 3.2. Theorem . (A\B) = ((A\C)\(B\C)). Problems 4. Symmetri c functions, sums x\ H f-xf , an d BernoulU numbers 1 9 Determinant relations between ok (x\,..., x n), J * (X\ ,..., x„ ) = x f H hx j and p^ (x\, ..., x„ ) = l l n ^2 x\ ... x n . A determinant formula for S„ (k) = 1 " -\ h (k — 1)". The Bernoulli numbers and S n(k). 4.4. Theorem . Let u = S\(x) andv — S2W. Then for k > 1 there exist polynomials p^ and q^ such that S2k+\ W = u 2pjc(u) and S^M = v 4k(u)- Problems Solutions Chapter II. Linea r spaces 3 5 Historical remarks: Hamilton and Grassmann 5. The dual space. The orthogonal complement 3 7 Linear equations and their application to the following theorem: Vlll CONTENTS 5.4.3. Theorem . If a rectangle with sides a andb is arbitrarily cut into squares with sides x\,... ,x„ then — € Q and — £ Qfor all i. a b Problems 6. Th e kernel (null space) and the image (range) o f an operator. Th e quotient space 6.2.1. Theorem . Ker^* = (ImA) 1- andlmA* = (KQTA^. Fredholm's alternative . Kronecker-Capelli' s theorem . Criteri a fo r solvabilit y o f th e matri x equation C = AXB.
Load more

Recommended publications
  • 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]).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 1111: Linear Algebra I
    1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 11 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 11 1 / 13 Previously on. Theorem. Let A be an n × n-matrix, and b a vector with n entries. The following statements are equivalent: (a) the homogeneous system Ax = 0 has only the trivial solution x = 0; (b) the reduced row echelon form of A is In; (c) det(A) 6= 0; (d) the matrix A is invertible; (e) the system Ax = b has exactly one solution. A very important consequence (finite dimensional Fredholm alternative): For an n × n-matrix A, the system Ax = b either has exactly one solution for every b, or has infinitely many solutions for some choices of b and no solutions for some other choices. In particular, to prove that Ax = b has solutions for every b, it is enough to prove that Ax = 0 has only the trivial solution. Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 11 2 / 13 An example for the Fredholm alternative Let us consider the following question: Given some numbers in the first row, the last row, the first column, and the last column of an n × n-matrix, is it possible to fill the numbers in all the remaining slots in a way that each of them is the average of its 4 neighbours? This is the \discrete Dirichlet problem", a finite grid approximation to many foundational questions of mathematical physics. Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 11 3 / 13 An example for the Fredholm alternative For instance, for n = 4 we may face the following problem: find a; b; c; d to put in the matrix 0 4 3 0 1:51 B 1 a b -1C B C @0:5 c d 2 A 2:1 4 2 1 so that 1 a = 4 (3 + 1 + b + c); 1 8b = 4 (a + 0 - 1 + d); >c = 1 (a + 0:5 + d + 4); > 4 < 1 d = 4 (b + c + 2 + 2): > > This is a system with 4:> equations and 4 unknowns.
    [Show full text]
  • 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.
    [Show full text]
  • Polynomials and Hankel Matrices
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Polynomials and Hankel Matrices Miroslav Fiedler Czechoslovak Academy of Sciences Institute of Mathematics iitnci 25 115 67 Praha 1, Czechoslovakia Submitted by V. Ptak ABSTRACT Compatibility of a Hankel n X n matrix W and a polynomial f of degree m, m < n, is defined. If m = n, compatibility means that HC’ = CfH where Cf is the companion matrix of f With a suitable generalization of Cr, this theorem is gener- alized to the case that m < n. INTRODUCTION By a Hankel matrix [S] we shall mean a square complex matrix which has, if of order n, the form ( ai +k), i, k = 0,. , n - 1. If H = (~y~+~) is a singular n X n Hankel matrix, the H-polynomial (Pi of H was defined 131 as the greatest common divisor of the determinants of all (r + 1) x (r + 1) submatrices~of the matrix where r is the rank of H. In other words, (Pi is that polynomial for which the nX(n+l)matrix I, 0 0 0 %fb) 0 i 0 0 0 1 LINEAR ALGEBRA AND ITS APPLICATIONS 66:235-248(1985) 235 ‘F’Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017 0024.3795/85/$3.30 236 MIROSLAV FIEDLER is the Smith normal form [6] of H,. It has also been shown [3] that qN is a (nonzero) polynomial of degree at most T. It is known [4] that to a nonsingular n X n Hankel matrix H = ((Y~+~)a linear pencil of polynomials of degree at most n can be assigned as follows: f(x) = fo + f,x + .
    [Show full text]
  • Matrix Lie Groups
    Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2).
    [Show full text]
  • Inverse Problems for Hankel and Toeplitz Matrices
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Inverse Problems for Hankel and Toeplitz Matrices Georg Heinig Vniversitiit Leipzig Sektion Mathematik Leipzig 7010, Germany Submitted by Peter Lancaster ABSTRACT The problem is considered how to construct a Toeplitz or Hankel matrix A from one or two equations of the form Au = g. The general solution is described explicitly. Special cases are inverse spectral problems for Hankel and Toeplitz matrices and problems from the theory of orthogonal polynomials. INTRODUCTION When we speak of inverse problems we have in mind the following type of problems: Given vectors uj E C”, gj E C”’ (j = l,.. .,T), we ask for an m x n matrix A of a certain matrix class such that Auj = gj (j=l,...,?-). (0.1) In the present paper we deal with inverse problems with the additional condition that A is a Hankel matrix [ si +j] or a Toeplitz matrix [ ci _j]. Inverse problems for Hankel and Toeplitz matices occur, for example, in the theory of orthogonal polynomials when a measure p on the real line or the unit circle is wanted such that given polynomials are orthogonal with respect to this measure. The moment matrix of p is just the solution of a certain inverse problem and is Hankel (in the real line case) or Toeplitz (in the unit circle case); here the gj are unit vectors. LINEAR ALGEBRA AND ITS APPLICATIONS 165:1-23 (1992) 1 0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of tbe Americas, New York, NY 10010 0024-3795/92/$5.00 2 GEORG HEINIG Inverse problems for Toeplitz matrices were considered for the first time in the paper [lo]of M.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [Show full text]