DOCSLIB.ORG

Constructing Copositive Matrices from Interior Matrices∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constructing Copositive Matrices from Interior Matrices∗ Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp. 9-20, January 2008 ELA CONSTRUCTING COPOSITIVE MATRICES FROM INTERIOR MATRICES∗ CHARLES R. JOHNSON† AND ROBERT REAMS‡ Abstract. Let A be an n by n symmetric matrix with real entries. Using the l1-norm for vectors + n and letting S1 = {x ∈ R |||x||1 =1,x ≥ 0},thematrixA is said to be interior if the quadratic form T + x Ax achieves its minimum on S1 in the interior. Necessary and sufficient conditions are provided for a matrix to be interior. A copositive matrix is referred to as being exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. A method is provided for constructing exceptional copositive matrices by completing a partial copositive matrix that has certain specified overlapping copositive interior principal submatrices. Key words. Copositive matrix, Interior matrix, Quadratic form, Almost positive semidefinite, Exceptional copositive matrix, Extreme copositive matrix. AMS subject classifications. 15A18, 15A48, 15A57, 15A63. T n 1. Introduction. We will call a vector v =(v1,...,vn) ∈ R nonnegative, n×n denoted v ≥ 0, if vi ≥ 0, for all i,1≤ i ≤ n. Similarly, a matrix A ∈ R will be called nonnegative, in the event that all its entries are nonnegative. A symmetric matrix A ∈ Rn×n will be called positive semidefinite (positive definite)ifxT Ax ≥ 0 for all x ∈ Rn (xT Ax > 0 for all x ∈ Rn,x= 0). A symmetric matrix A ∈ Rn×n will be called copositive (strictly copositive)ifxT Ax ≥ 0 for all x ∈ Rn,x≥ 0(xT Ax > 0 for all x ∈ Rn,x ≥ 0,x = 0). The vector in Rn of all ones will be denoted by e,so T n n×n that e =(1, 1,...,1) ∈ R .ThematrixEij ∈ R denotes the matrix with a 1 in n the (i, j) position and zeroes elsewhere, while ei ∈ R denotes the vector with a 1 in the ith position and zeroes elsewhere. A symmetric matrix A ∈ Rn×n will be called almost positive semidefinite (almost positive definite)ifxT Ax ≥ 0 for all x ∈ Rn such that xT e =0(xT Ax > 0 for all x ∈ Rn, x = 0, such that xT e = 0)[13]. Note that if a matrix is copositive then all its principal submatrices are copositive. We will say that a matrix is partial copositive if it has some entries that are specified, and some that are not, and for every principal submatrix which has all its entries specified, this submatrix is copositive. It is easy to show that a 2 × 2 copositive matrix is ∗Received by the editors on August 8, 2007. Accepted for publication on January 9, 2008 Handling Editor: Daniel Hershkowitz. †Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187, USA ([email protected]). ‡Department of Mathematics, Virginia Commonwealth University, 1001 West Main Street, Rich- mond, VA 23284, USA ([email protected]). 9 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp. 9-20, January 2008 ELA 10 C.R. Johnson and R. Reams either positive semidefinite or nonnegative. Diananda [7] showed that every 3 × 3or 4 × 4 copositive matrix is the sum of a positive semidefinite matrix and a nonnegative matrix. We will call a copositive matrix exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. Thus, n × n exceptional copositive matrices can only occur for n ≥ 5. Horn [8] gave an example of an exceptional matrix (see Section 3 of this paper). By an extreme copositive matrix Q we mean that if Q = Q1 + Q2,whereQ1 and Q2 are copositive then Q1 = aQ,andQ2 =(1− a)Q, for some a such that 0 ≤ a ≤ 1. n + We will denote the vector l1-norm [12] by ||x||1 = i=1 |xi|,andbyS1 the n portion of the unit 1-sphere S1 = {x ∈ R |||x||1 =1} in the nonnegative orthant, + n n×n thus S1 = {x ∈ R |||x||1 =1,x ≥ 0}.LetA ∈ R be symmetric. The set + T S1 is compact so a minimum for x Ax is achieved on this set. We will say that A T is interior if such a minimum, namely min + x Ax, is achieved in the interior of x∈S1 + S1 .NotethatA is not necessarily copositive in this definition, although A being copositive is our primary interest. This definition does not preclude the possibility that the minimum is also achieved on the boundary, just that it will be achieved in the interior in any case. For example, with n =3and 1 −11 A = −11−1 , 1 −11 the matrix A is interior, since the minimum of the quadratic form xT Ax is achieved + 1 T in the interior of S1 at the vector 4 (1, 2, 1) . Although the minimum is also achieved 1 T n×n on the boundary at 2 (1, 1, 0) . For a symmetric matrix A ∈ R , if the minimum T + of x Ax is not achieved in the interior of S1 then the minimum is achieved at a vector u with some zero components. Let us call u ∈ Rk,where1≤ k<n,the vector consisting of the positive components of u,sothatu is of the form u = u ,...,u T S+ {x ∈ Rk|||x|| ,x ≥ } ( i1 ik ) , and is in the interior of 1 = p =1 0 ,and where i1,...,ik are determined by deleting the zero components of the minimizer u. T T Then minx∈S+ x Ax =min + x A x ,whereA is the principal submatrix of A 1 x ∈S1 obtained by deleting the ith row and column of A, for every i such that i ∈{i1,...,ik}. Thus A is an interior matrix. To be consistent with when n = 1 we will say that the matrix A =(a)isinterior, + T even though S1 = {1} has no interior or boundary. In this case, min + x Ax = x∈S1 2 minx=1 ax = a. Note that for a 1 × 1matrixA, being copositive and not strictly copositive implies that A = (0). For another example, this time with n =2,take 21 T T 2 2 A = .Thenwithx =[x1 x2 ] we have x Ax = x +(x1 + x2) ,so 11 1 T T min + x Ax = 1 and the minimum is achieved at u =[0 1] (in the notation of x∈S1 the preceding paragraph) and A = (1) is an interior matrix and u =1. Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp. 9-20, January 2008 ELA Constructing Copositive Matrices from Interior Matrices 11 In Section 2 we provide necessary and sufficient conditions for a matrix to be interior. In Section 3 we employ interior principal submatrices to construct coposi- tive matrices that are exceptional. In particular, we will show how to construct an exceptional copositive matrix by performing a completion of a matrix that has certain overlapping principal submatrices specified, which are copositive and interior. Section 4 contains examples that demonstrate this method of construction. + 2. Interior matrices. Theorem 1 below minimizes the quadratic form on S1 . + A variation of Theorem 1 in which the quadratic form is maximized on S1 ,and xT Ax has no maxima on the boundary, was proved by Sidorenko in [17], whose proof relies on [5] and [6]. Our interest lies in minimizing the quadratic form because with copositive matrices, which are not strictly copositive, the value of that minimum on the nonnegative orthant is zero. We are also interested in the location of the minimizer, which may move from the boundary to the interior, or vice versa, on changing from maximizing to minimizing. Our proofs are self-contained and different than Sidorenko’s. Theorem 1. Let A ∈ Rn×n be symmetric. Then A is interior if and only if (i) and (ii) hold. (i) There exists u>0, ||u||1 =1,andµ ∈ R such that Au = µe; (ii) yT Ay ≥ 0 for all y ∈ Rn such that yT e =0(i.e. A is almost positive semidefinite). T T If A is interior then µ =min + x Ax = u Au. x∈S1 If (i) holds and yT Ay > 0 for all y =0 such that yT e =0(i.e. A is almost positive + definite), then u>0 is the only minimizer in S1 . + If A is interior and u is not the only minimizer in S1 then A is singular, and Ay =0 for some y =0 such that yT e =0. Proof. Suppose that A is interior. Since there is a minimizer u for xT Ax in S+ xT Ax the interior of 1 we may use Lagrange Multipliers and minimize subject to T n ||x||1 =1.Thus∇(x Ax)=λ∇( i=1 xi), when x = u, which implies 2(Au)i = λ, λ T T or Au = µe,whereµ = . So (i) holds. Evidently, µ =min + x Ax = u Au. 2 x∈S1 Let y ∈ Rn be any y =0besuchthat yT e =0.Thenforsomec =0, u + cy ≥ 0, (u+cy)T (u+cy) T u A ≥ u Au ||u cy||1 and since is a minimizer we have ||u+cy||1 ||u+cy||1 .But + = (u + cy)T e =1,sowehaveuT Au +2cyT Au + c2yT Ay ≥ uT Au,orc2yT Ay ≥ 0, proving (ii). + Conversely, suppose (i) and (ii) hold. Let z be any vector in S1 . Since the vector u together with a basis for {y|yT e =0} spans Rn we can write z = νu + y,where Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp.
Load more

Recommended publications
  • Some Aspects of the Optimization Over the Copositive and the Completely Positive Cone
    Some aspects of the optimization over the copositive and the completely positive cone Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) Dem Fachbereich IV der Universit¨at Trier vorgelegt von Julia Witzel Trier, 2013 Eingereicht am 17.07.2013 Gutachter: Prof. Dr. Mirjam Dur¨ Prof. Dr. Gabriele Eichfelder Tag der mundlichen¨ Prufung:¨ 25.09.2013 Die Autorin hat im Laufe des Promotionsverfahrens geheiratet und hieß zu dessen Beginn Julia Sponsel. Zusammenfassung Optimierungsprobleme tauchen in vielen praktischen Anwendungsbereichen auf. Zu den einfachsten Problemen geh¨oren lineare Optimierungsprobleme. Oft reichen lineare Probleme allerdings zur Modellierung praktischer Anwendun- gen nicht aus, sondern es muss die Ganzzahligkeit von Variablen gefordert werden oder es tauchen nichtlineare Nebenbedingungen oder eine nichtlineare Zielfunktion auf. Eine wichtige Problemklasse stellen hierbei quadratische Pro- bleme dar, welche im Allgemeinen aber schwer zu l¨osen sind. Ein Ansatz, der sich in den letzten Jahren entwickelt und verbreitet hat, besteht in der Umfor- mulierung quadratischer Probleme in sogenannte kopositive Programme. Ko- positive Programme sind lineare Optimierungsprobleme uber¨ dem Kegel der kopositiven Matrizen n×n T T n Cn = fA 2 R : A = A ; x Ax ≥ 0 fur¨ alle x 2 R ; x ≥ 0g; oder dem Dualkegel, dem Kegel der vollst¨andig positiven Matrizen m ∗ n X T n o Cn = xixi : xi 2 R ; xi ≥ 0 fur¨ alle i = 1; : : : ; m : i=1 Die Schwierigkeit dieser Probleme liegt in der Kegelbedingung. Aber nicht nur das Optimieren uber¨ diesen Kegeln ist schwierig. Wie von Murty und Kabadi (1987) und Dickinson und Gijben (2013) gezeigt ist es NP-schwer zu testen, ob eine gegebene Matrix kopositiv oder vollst¨andig positiv ist.
    [Show full text]
  • Copositive Plus Matrices
    Copositive Plus Matrices Willemieke van Vliet Master Thesis in Applied Mathematics October 2011 Copositive Plus Matrices Summary In this report we discuss the set of copositive plus matrices and their properties. We examine certain subsets of copositive plus matrices, copositive plus matrices with small dimensions, and the copositive plus cone and its dual. Furthermore, we consider the Copositive Plus Completion Problem, which is the problem of deciding whether a matrix with unspecified entries can be completed to obtain a copositive plus matrix. The set of copositive plus matrices is important for Lemke's algorithm, which is an al- gorithm for solving the Linear Complementarity Problem (LCP). The LCP is the problem of deciding whether a solution for a specific system of equations exists and finding such a solution. Lemke's algorithm always terminates in a finite number of steps, but for some prob- lems Lemke's algorithm terminates with no solution while the problem does have a solution. However, when the data matrix of the LCP is copositive plus, Lemke's algorithm always gives a solution if such solution exists. Master Thesis in Applied Mathematics Author: Willemieke van Vliet First supervisor: Dr. Mirjam E. D¨ur Second supervisor: Prof. dr. Harry L. Trentelman Date: October 2011 Johann Bernoulli Institute of Mathematics and Computer Science P.O. Box 407 9700 AK Groningen The Netherlands Contents 1 Introduction 1 1.1 Structure . .1 1.2 Notation . .1 2 Copositive Plus Matrices and their Properties 3 2.1 The Class of Copositive Matrices . .3 2.2 Properties of Copositive Matrices . .4 2.3 Properties of Copositive Plus Matrices .
    [Show full text]
  • Cambridge University Press 978-1-108-47871-7 — Matrix Positivity Charles R
    Cambridge University Press 978-1-108-47871-7 — Matrix Positivity Charles R. Johnson , Ronald L. Smith , Michael J. Tsatsomeros Index More Information Index p-Newton, 159 completion of a partial matrix, 153 t-th Hadamard power, 128 completion to a P-matrix, 88 (convex) cone, 162 cone,3,39 (i,j)-path, 157 congruence of A, 168 (right) Perron eigenvector, 58 conjugate transpose, 1 (right) null space, 114 contiguous minors, 9 (strictly) codefinite pair, 172 convex, 3 (strictly) copositive of order m, 171 convex combination, 3 convex hull, 4 acute cone, 40 Copositive, 164 allow MSP,30 copositive, 10 allow RSP,30 copositive +,10 allow SP,25 copositive with respect to a cone K, 167 almost (strictly) semimonotone, 53 Copositive-plus, 164 almost C, 177 CP-rank, 11 almost C+, 177 critical exponent, 135 almost SC, 177 cycle of length, 59 almost diagonally dominant P-matrices, 88 cycle product, 120 almost principal submatrices, 109 cycle product inequalities, 120 associated quadratic form, 174 determinantal inequalities, 8 B-matrix, 58 diagonal Lyapunov solution, 91 block graph, 155 diagonal Lyapunov solutions, 95 block P-matrices, 88 diagonally dominant matrix, 2 buckle, 157 diagonally dominant of its column entries, 94 diagonally dominant of its columns, 93 Cayley transform, 57 diagonally dominant of its row entries, 94 Cholesky factorization, 10 diagonally dominant of its rows, 93 column (row) deleted submatrix, 16 diagonally symmetrizable, 127 column diagonally dominant, 12 directed graph, 59 common semipositivity vector, 59 directed graph of A, 121 comparison matrix, 58 dispersion, 75 complementary nullities, 114 doubly nonnegative matrices, 10, 135 completely positive matrix, 11 doubly positive matrices, 10 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47871-7 — Matrix Positivity Charles R.
    [Show full text]
  • The Copositive-Plus Matrix Completion Problem, Xuhua Liu
    Alabama Journal of Mathematics ISSN 2373-0404 39 (2015) The Copositive-Plus Matrix Completion Problem Xuhua Liu Ronald L. Smith Department of Mathematics Department of Mathematics The University of Tennessee at Chattanooga The University of Tennessee at Chattanooga Chattanooga, TN 37403, USA Chattanooga, TN 37403, USA We derive a useful characterization of 3-by-3 copositive-plus matrices in terms of their entries, and use it to solve the copositive-plus matrix completion problem in the case of specified diagonal by showing that a partial copositive-plus matrix with graph G has a copositive-plus completion if and only if G is a pairwise disjoint union of complete subgraphs. Introduction (cf. Hiriart-Urruty and Seeger (2010)). See recent surveys Hiriart-Urruty and Seeger (2010) and Ikramov and Savel’eva In this article, the superscript “>” denotes transposi- n (2000) on copositive matrices and references therein. tion and R the set of all real n-vectors. A vector x = A partial matrix is one in which some entries are spec- > 2 n (x1;:::; xn) R is said to be nonnegative, denoted by ified, while the remaining entries are unspecified and free x > 0, if xi > 0 for all i = 1;:::; n. to be chosen. A completion of a partial matrix is a choice × Let S n denote the set of all n n real symmetric matrices. of values for the unspecified entries. A matrix completion 2 A matrix A S n is said to be problem asks which partial matrices have completions with + ∗ > n a desired property. A partial C (C, C , resp.) matrix is a (1) (real) positive semidefinite if x Ax > 0 for all x 2 R ; real symmetric partial matrix such that every fully specified > + ∗ ∗ (2) copositive if x Ax > 0 for all x > 0; principal submatrix is C (C, C , resp.).
    [Show full text]
  • Copositive Matrices with Circulant Zero Support
    Copositive matrices with circulant zero support set Roland Hildebrand ∗ July 19, 2021 Abstract 1 n Let n ≥ 5 and let u ,...,u be nonnegative real n-vectors such that the indices of their positive elements form the sets {1, 2,...,n − 2}, {2, 3,...,n − 1},..., {n, 1,...,n − 3}, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms 1 n n which vanish on the vectors u ,...,u is a face of the copositive cone C . We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite forms, and study their properties. If the vectors u1,...,un and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we say it has minimal circulant zero support set, and otherwise non-minimal circulant zero support set. We show that forms with non-minimal circulant zero support set are always extremal, and forms with minimal circulant zero support sets can be extremal only if n is odd. We construct explicit examples of extremal forms with non-minimal circulant zero support set for any order n ≥ 5, and examples of extremal forms with minimal circulant zero support set for any odd order n ≥ 5. The set of all forms with non-minimal circulant zero support set, i.e., defined by different collections u1,...,un of zeros, is a submanifold of codimension 2n, the set of all forms with minimal circulant zero support set a submanifold of codimension n.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Copositivity And
    Mathematisches Forschungsinstitut Oberwolfach Report No. 52/2017 DOI: 10.4171/OWR/2017/52 Copositivity and Complete Positivity Organised by Abraham Berman, Haifa Immanuel M. Bomze, Vienna Mirjam D¨ur, Augsburg Naomi Shaked-Monderer, Yezreel Valley 29 October – 4 November 2017 Abstract. A real matrix A is called copositive if xT Ax ≥ 0 holds for all Rn x ∈ +. A matrix A is called completely positive if it can be factorized as A = BBT , where B is an entrywise nonnegative matrix. The concept of copositivity can be traced back to Theodore Motzkin in 1952, and that of complete positivity to Marshal Hall Jr. in 1958. The two classes are related, and both have received considerable attention in the linear algebra community and in the last two decades also in the mathematical optimization commu- nity. These matrix classes have important applications in various fields, in which they arise naturally, including mathematical modeling, optimization, dynamical systems and statistics. More applications constantly arise. The workshop brought together people working in various disciplines re- lated to copositivity and complete positivity, in order to discuss these con- cepts from different viewpoints and to join forces to better understand these difficult but fascinating classes of matrices. Mathematics Subject Classification (2010): 15B48, 15A23, 15A63, 37B25, 37C75, 90C20, 90C22, 93D09 . Introduction by the Organisers T Rn A real matrix A is called copositive if x Ax 0 holds for all x +. Obviously, every positive semidefinite matrix is copositive,≥ and so is every entry∈ wise symmet- ric nonnegative matrix. However, for n 5 the cone of n n copositive matrices is considerably larger and more complex than≥ both the semidefinite× and symmetric nonnegative matrix cones.
    [Show full text]
  • Multilinear Algebra
    A Complete Bibliography of Publications in Linear [and] Multilinear Algebra Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 18 May 2021 Version 1.15 Title word cross-reference (0; 1) [FV97, Kim97, Zel85, BF15, BS18d, Cao19, KS83, LN10, Min74d, Zho16]. (0; 1; −1) [GZ99]. (1; −1) [KS84]. (1; 0) [MP13b]. (1; 1) [Zha20a, Che03b]. (1; 2; 3) [GK18]. (2) [Che95a]. (2; 2; 2) [SF17]. (2;p) [YD14]. (α, β)[YZ13]. (α, β; γ) [KO10]. (AXB; CXD)=(E;F) [YLW15]. (AXB; CXD)=(E;G) + [YTL20]. (b; c)[SCK´ 20, Boa19, Dra18b, KCICV18]. (e(xrys)) [Pre85]. (k) [Cha77, Cha79a]. (k + 1) [BT06]. (k; l)[Rem90].(k =2; 3; 4) [Sta15b]. (m; C) [MCL20]. (m; m − 1; 0) [HZ21]. (max; +) [HKT15]. (n + 1) [Sch78, Sch75]. (n; k)=(4; 2) [JLL88]. (n; m) [CM19]. (p; h) [DV18]. (p; q) [KL95]. (p; σ) [ADRP14].p (qr)[Raw81].(R; S) [fLLyP15, Pen15]. (s; p) − w [Ras17]. s ( 5 − 1)=2[Sim95].(T ) [Che95b]. (υ, k; λ)[Der83].(Zm) [How86]. −2 [Ano16b, BPS16]. 0 [HNS99, HLZ05, HL20, JS96, Wu09]. 0; 1 [MS79, BX07]. 0 62 F (Am) [JL90]. 1 [CM15, DFK19, Fra17, HNS99, HLZ05, HL20, JS96, KP14a, Min77, Sta10, Wu09]. 1; 2; 1[KHG13].f1; 2; 3g [ZX10]. f1; 2; 4g 1 [ZX10]. f1; 3g [CIM19]. 10 [LN10]. 12 [EM95].
    [Show full text]
  • Matrix Theory
    Matrix Theory Xingzhi Zhan +VEHYEXI7XYHMIW MR1EXLIQEXMGW :SPYQI %QIVMGER1EXLIQEXMGEP7SGMIX] Matrix Theory https://doi.org/10.1090//gsm/147 Matrix Theory Xingzhi Zhan Graduate Studies in Mathematics Volume 147 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15-01, 15A18, 15A21, 15A60, 15A83, 15A99, 15B35, 05B20, 47A63. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 Library of Congress Cataloging-in-Publication Data Zhan, Xingzhi, 1965– Matrix theory / Xingzhi Zhan. pages cm — (Graduate studies in mathematics ; volume 147) Includes bibliographical references and index. ISBN 978-0-8218-9491-0 (alk. paper) 1. Matrices. 2. Algebras, Linear. I. Title. QA188.Z43 2013 512.9434—dc23 2013001353 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government.
    [Show full text]
  • Complementarity Properties of Peirce-Diagonalizable Linear Transformations on Euclidean Jordan Algebras
    Optimization Methods and Software Vol. 00, No. 00, January 2009, 1{14 Complementarity properties of Peirce-diagonalizable linear transformations on Euclidean Jordan algebras M. Seetharama Gowdaa∗, J. Taob, and Roman Sznajderc aDepartment of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA ; bDepartment of Mathematics and Statistics, Loyola University Maryland, Baltimore, Maryland 21210, USA; c Department of Mathematics, Bowie State University, Bowie, Maryland 20715, USA (xxxxxx) Peirce-diagonalizable linear transformations on a Euclidean Jordan algebra are of the form P P L(x) = A•x := aij xij , where A = [aij ] is a real symmetric matrix and xij is the Peirce decomposition of an element x in the algebra with respect to a Jordan frame. Examples of such transformations include Lyapunov transformations and quadratic representations on Euclidean Jordan algebras. Schur (or Hadamard) product of symmetric matrices provides another example. Motivated by a recent generalization of the Schur product theorem, in this article, we study general and complementarity properties of such transformations. Keywords: Peirce-diagonalizable transformation, linear complementarity problem, Euclidean Jordan algebra, symmetric cone, Schur/Hadamard product, Lypaunov transformation, quadratic representation. AMS Subject Classification: 15A33; 17C20; 17C65; 90C33. 1. Introduction Let L be a linear transformation on a Euclidean Jordan algebra (V; ◦; h·; ·i) of rank r and let Sr denote the set of all real r × r symmetric matrices. We say that L is Peirce-diagonalizable if there exist a Jordan frame fe1; e2; : : : ; erg (with a specified r ordering of its elements) in V and a matrix A = [aij] 2 S such that for any x 2 V P with its Peirce decomposition x = i≤j xij with respect to fe1; e2; : : : ; erg, we have X L(x) = A • x := aijxij: (1) 1≤i≤j≤r The above expression defines a Peirce-diagonal transformation and a Peirce diag- onal representation of L.
    [Show full text]
  • Perron–Frobenius Property of Copositive Matrices, and a Block Copositivity Criterion
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Linear Algebra and its Applications 429 (2008) 68–71 www.elsevier.com/locate/laa Perron–Frobenius property of copositive matrices, and a block copositivity criterion Immanuel M. Bomze Department of Statistics and Decision Support Systems, University of Vienna, Austria Received 21 November 2007; accepted 5 February 2008 Available online 25 March 2008 Submitted by R.A. Brualdi Abstract Haynsworth and Hoffman proved in 1969 that the spectral radius of a symmetric copositive matrix is an eigenvalue of this matrix. This note investigates conditions which guarantee that an eigenvector correspond- ing to this dominant eigenvalue has no negative coordinates, i.e., whether the Perron–Frobenius property holds. Also a block copositivity criterion using the Schur complement is specified which may be helpful to reduce dimension in copositivity checks and which generalizes results proposed by Andersson et al. in 1995, and Johnson and Reams in 2005. Apparently, the latter five researchers were unaware of the more general results by the author precedingly published in 1987 and 1996, respectively. © 2008 Published by Elsevier Inc. Keywords: Dominant eigenvalue; Positive eigenvector; Schur complement; Spectral radius The Perron–Frobenius property has many well-studied consequences, also for matrices with some negative entries, see [6–9] and references therein. In [4], Haynsworth and Hoffman prove, among other results and in a more general setting, that a symmetric copositive matrix has a dominant (positive) eigenvalue. This way, part of the Perron–Frobenius theorem is extended to a class of matrices larger than the nonnegative ones, as copositive matrices also may have some negative entries, even if they are positive-semidefinite.
    [Show full text]
  • Nearly Positive Matrices Bryan Shader, Naomi Shaked-Monderer and Daniel B
    Nearly positive matrices Bryan Shader, Naomi Shaked-Monderer and Daniel B. Szyld Research Report 13-07-17 July 2013 Department of Mathematics Temple University This report is available in the World Wide Web at http://www.math.temple.edu/szyld Nearly positive matrices∗ Bryan Shadery Naomi Shaked-Mondererz Daniel B. Szyldx Abstract Nearly positive (NP) matrices are nonnegative matrices which, when premultiplied by orthogonal matrices as close to the identity as one wishes, become positive. In other words, all columns of an NP matrix are mapped simultaneously to the interior of the nonnegative cone by mutiplication by a sequence of orthogonal matrices converging to the identity. In this paper, NP matrices are analyzed and characterized in several cases. Different necessary and sufficient conditions for a nonnegative matrix to be an NP matrix are presented. A connection to completely positive matrices is also presented. Keywords: Nonnegative matrices, Positive matrices, Completely positive matrices. AMS subject classification: 15F10 1 Introduction Consider the cone of nonnegative vectors in the m-dimensional space m m R+ = fx 2 R ; x ≥ 0g. Any nonzero vector v in its boundary is nonnega- tive, but not positive. It is not hard to see that an infinitesimal rotation in the m appropriate direction can bring this vector into the interior of the cone R+ . In other words, one can build a sequence of orthogonal matrices Q(`) such that lim`!1 Q(`) = I with the property that Q(`)v > 0. For two non-orthogonal nonnegative vectors u, v, one can also build a se- quence of orthogonal matrices, such that both Q(`)u > 0 and Q(`)v > 0 [6, Theorem 6.12].
    [Show full text]
  • Copositive Programming – a Survey
    Copositive Programming { a Survey Mirjam D¨ur Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands. [email protected] Copositive programming is a relatively young field in mathematical opti- mization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorith- mic solution approaches for copositive programs. 1 Introduction A copositive program is a linear optimization problem in matrix variables of the following form: min hC; Xi s: t: hAi;Xi = bi (i = 1; : : : ; m); (1) X 2 C; where C is the cone of so-called copositive matrices, that is, the matrices whose n quadratic form takes nonnegative values on the nonnegative orthant R+: T n C = fA 2 S : x Ax ≥ 0 for all x 2 R+g (here S is the set of symmetric n × n matrices, and the inner product of two Pn matrices in (1) is hA; Bi := trace(BA) = i;j=1 aijbij, as usual). Obviously, every positive semidefinite matrix is copositive, and so is every entrywise nonnegative matrix, but the copositive cone is significantly larger than both the semidefinite and the nonnegative matrix cones.
    [Show full text]