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Complementarity Properties of Peirce-Diagonalizable Linear Transformations on Euclidean Jordan Algebras

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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. Our first example is obtained by taking V = Sr with the canonical Jordan frame (see Section 2 for various notations and definitions). In this case, A • x reduces to the well-known Schur (also known as Hadamard) product of two symmetric matrices and L is the corresponding induced transformation. On a general Euclidean Jordan algebra, two basic examples are: The Lyapunov transformation La and the quadratic representation Pa corresponding to any ele- ∗Corresponding author. Email: [email protected] ISSN: 1055-6788 print/ISSN 1029-4937 online c 2009 Taylor & Francis DOI: 10.1080/1055678xxxxxxxxxxxxx http://www.informaworld.com 2 Taylor & Francis and I.T. Consultant ment a 2 V . These are, respectively, defined on V by 2 La(x) = a ◦ x and Pa(x) = 2a ◦ (a ◦ x) − a ◦ x: If the spectral decomposition of a is given by a = a1e1 + a2e2 + ··· + arer; where fe1; e2; : : : ; erg is a Jordan frame and a1; a2; : : : ; ar are the eigenvalues of a, then with respect to this Jordan frame and the induced Peirce decomposition P x = i≤j xij of any element, we have [20] X ai + aj X L (x) = ( )x and P (x) = (a a )x : (2) a 2 ij a i j ij i≤j i≤j Thus, both La and Pa have the form (1) and the corresponding matrices are given, ai+aj respectively, by [ 2 ] and [aiaj]. We shall see (in Section 3) that every transfor- mation L given by (1) is a linear combination of quadratic representations. Our fourth example deals with L¨ownerfunctions. Given a differentiable function φ : R!R, consider the corresponding L¨ownerfunction Φ : V ! V defined by (the spectral decompositions) a = λ1e1 + λ2e2 + ··· + λrer and Φ(a) = φ(λ1)e1 + φ(λ2)e2 + ··· + φ(λr)er: P P Then the directional derivative of Φ at a = λiei in the direction of x = xij (Peirce decomposition written with respect to fe1; e2; : : : ; erg) is given by 0 X Φ (a; x) := aijxij; i≤j φ(λi)−φ(λj ) where aij := (which, by convention, is the derivative of φ when λi = λj). λi−λj We now note that (for a fixed a), Φ0(a; x) is of the form (1). In [16], Kor´anyi studies the operator monotonicity of Φ based on such expressions. Expressions like (1) also appear in connection with the so-called uniform non- singularity property, see the recent paper [2] for further details. Our objective in this paper is to study Peirce-diagonalizable transformations, and, in particular, describe their complementarity properties. Consider a Euclidean Jordan algebra V with the corresponding symmetric cone K. Given a linear trans- formation L on V and an element q 2 V , the (symmetric cone) linear complemen- tarity problem, denoted by LCP(L; K; q), is to find an x 2 V such that x 2 K; L(x) + q 2 K; and hL(x) + q; xi = 0: (3) This problem is a generalization of the standard linear complementarity problem n×n n [3] which corresponds to a square matrix M 2 R , the cone R+, and a vector q 2 Rn; it is a special case of a variational inequality problem [4]. In the theory of complementarity problems, global uniqueness and solvability is- sues are of fundamental importance. A linear transformation L is said to have the Q-property (GUS-property) if for every q 2 V , LCP(L; K; q) has a solution (re- spectively, unique solution); it is said to have the R0-property if LCP(L; K; 0) has a unique solution (namely zero). In the context of La, GUS, Q, and R0 properties are all equivalent; moreover, they hold if and only if a > 0, that is, when a belongs Optimization Methods and Software 3 to the interior of symmetric cone of V . Similarly, for Pa, the properties GUS, Q, and R0 properties are equivalent, see [8]. In this paper, we extend these results to certain Peirce-diagonalizable transformations. Our motivation for this study also comes from a recent result which generalizes the well-known Schur product the- orem to Euclidean Jordan algebras [19]: If A in (1) is positive semidefinite and x 2 K, then A • x 2 K. The organization of the paper is as follows: Section 2 covers some basic material. In Section 3, we cover general properties of Peirce-diagonalizable transformations. Section 4 deals with the copositivity property. In this section, we introduce certain cones between the cones of completely positive matrices and doubly nonnegative matrices. Section 5 deals with the Z-property, and finally in Section 6, we cover the complementarity properties. 2. Some preliminaries Throughout this paper, we fix a Euclidean Jordan algebra (V; ◦; h·; ·i) of rank r with the corresponding symmetric cone (of squares) K [5], [9]. It is well-known that V is a product of simple algebras, each isomorphic to one of the matrix algebras Sn, Hn, Qn (which are the spaces of n × n Hermitian matrices over real numbers/complex numbers/quaternions), O3 (the space of 3 × 3 Hermitian matrices over octonions), or the Jordan spin algebra Ln (n ≥ 3). In the matrix algebras, the (canonical) inner product is given by hX; Y i = Re trace(XY ) and in Ln, it is the usual inner product on Rn. The symmetric cone of Sn will be n denoted by S+ (with a similar notation in other spaces). In any matrix algebra, let Eij be the matrix with ones in the (i; j) and (j; i) slots and zeros elsewhere; we write, Ei := Eii. In a matrix algebra of rank r, fE1;E2;:::;Erg is the canonical Jordan frame. We use the notation A = [aij] to say that A is a matrix with components aij; we also write r A 0 , A 2 S+ and x ≥ 0 (> 0) , x 2 K (int(K)): For a given Jordan frame fe1; e2; : : : ; erg in V , we consider the corresponding Peirce decomposition ([5], Theorem IV.2.1) X V = Vij; 1≤i≤j≤r 1 where Vii = R ei and Vij = fx 2 V : x ◦ ei = x = x ◦ ejg, when i 6= j. For any 2 P x 2 V , we get the corresponding Peirce decomposition x = i≤j xij. We note that this is an orthogonal direct sum and xii = xiei for all i, where xi 2 R. We recall a few new concepts and notations. Given a matrix A 2 Sr, a Euclidean Jordan algebra V and a Jordan frame fe1; e2; : : : ; erg with a specific ordering, we define the linear transformation DA : V ! V and Schur (or Hadamard) products in the following way: For any two elements x; y 2 V with Peirce decompositions P P x = i≤j xij and y = i≤j yij, X DA(x) = A • x := aijxij 2 V (4) 1≤i≤j≤r 4 Taylor & Francis and I.T. Consultant and r X 1 X x 4 y := hx ; y iE + hx ; y iE 2 Sr: (5) ii ii i 2 ij ij ij 1 1≤i<j≤r We immediately note that X 2 hDA(x); xi = hA • x; xi = aijjjxijjj = hA; x4 xi; (6) i≤j where the inner product on the far right is taken in Sr: When V is Sr with the canonical Jordan frame, the above products A • x and x 4 y coincide with the well-known Schur (or Hadamard) product of two symmetric matrices. This can be seen as follows. Consider matrices x = [αij] and y = [βij] in r P P S so that x = i≤j αijEij and y = i≤j βijEij are their corresponding Peirce de- compositions with respect to the canonical Jordan frame. Then simple calculations show that X A • x = aijαijEij = [aijαij] and x 4 y = [αijβij]; i≤j r where the latter expression is obtained by noting that in S , hEij;Eiji = 2 trace(Eij) = 2: It is interesting to note that (A • x)4 x = A • (x4 x); (7) where the `bullet' product on the right is taken in Sr with respect to the canonical Jordan frame.
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