Complementarity properties of Peirce-diagonalizable linear transformations on Euclidean Jordan algebras

Transcription

1 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 Gowda a, J. Tao b, and Roman Sznajder c a Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA ; b Department 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 L(x) = A x := a ij x ij, where A = [a ij ] is a real symmetric matrix and x ij 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; 90C Introduction Let L be a linear transformation on a Euclidean Jordan algebra (V,,, ) of rank r and let S r denote the set of all real r r symmetric matrices. We say that L is Peirce-diagonalizable if there exist a Jordan frame {e 1, e 2,..., e r } (with a specified ordering of its elements) in V and a matrix A = [a ij ] S r such that for any x V with its Peirce decomposition x = i j x ij with respect to {e 1, e 2,..., e r }, we have L(x) = A x := 1 i j r a ij x ij. (1) The above expression defines a Peirce-diagonal transformation and a Peirce diagonal representation of L. Our first example is obtained by taking V = S r 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 L a and the quadratic representation P a corresponding to any ele- Corresponding author. gowda@math.umbc.edu ISSN: print/issn online c 2009 Taylor & Francis DOI: / xxxxxxxxxxxxx

2 2 Taylor & Francis and I.T. Consultant ment a V. These are, respectively, defined on V by L a (x) = a x and P a (x) = 2a (a x) a 2 x. If the spectral decomposition of a is given by a = a 1 e 1 + a 2 e a r e r, where {e 1, e 2,..., e r } is a Jordan frame and a 1, a 2,..., a r are the eigenvalues of a, then with respect to this Jordan frame and the induced Peirce decomposition x = i j x ij of any element, we have [20] L a (x) = i j ( a i + a j )x ij and P a (x) = (a i a j )x ij. (2) 2 i j Thus, both L a and P a have the form (1) and the corresponding matrices are given, respectively, by [ ai+aj 2 ] and [a i a j ]. We shall see (in Section 3) that every transformation L given by (1) is a linear combination of quadratic representations. Our fourth example deals with Löwner functions. Given a differentiable function φ : R R, consider the corresponding Löwner function Φ : V V defined by (the spectral decompositions) a = λ 1 e 1 + λ 2 e λ r e r and Φ(a) = φ(λ 1 )e 1 + φ(λ 2 )e φ(λ r )e r. Then the directional derivative of Φ at a = λ i e i in the direction of x = x ij (Peirce decomposition written with respect to {e 1, e 2,..., e r }) is given by Φ (a; x) := i j a ij x ij, where a ij := φ(λi) φ(λj) λ i λ j (which, by convention, is the derivative of φ when λ i = λ j ). We now note that (for a fixed a), Φ (a; x) is of the form (1). In [16], Korányi studies the operator monotonicity of Φ based on such expressions. Expressions like (1) also appear in connection with the so-called uniform nonsingularity 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 transformation L on V and an element q V, the (symmetric cone) linear complementarity problem, denoted by LCP(L, K, q), is to find an x V such that x K, L(x) + q K, and L(x) + q, x = 0. (3) This problem is a generalization of the standard linear complementarity problem [3] which corresponds to a square matrix M R n n, the cone R n +, and a vector q R n ; it is a special case of a variational inequality problem [4]. In the theory of complementarity problems, global uniqueness and solvability issues are of fundamental importance. A linear transformation L is said to have the Q-property (GUS-property) if for every q V, LCP(L, K, q) has a solution (respectively, unique solution); it is said to have the R 0 -property if LCP(L, K, 0) has a unique solution (namely zero). In the context of L a, GUS, Q, and R 0 properties are all equivalent; moreover, they hold if and only if a > 0, that is, when a belongs

3 Optimization Methods and Software 3 to the interior of symmetric cone of V. Similarly, for P a, the properties GUS, Q, and R 0 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 theorem to Euclidean Jordan algebras [19]: If A in (1) is positive semidefinite and x K, then A x 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,,, ) 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 S n, H n, Q n (which are the spaces of n n Hermitian matrices over real numbers/complex numbers/quaternions), O 3 (the space of 3 3 Hermitian matrices over octonions), or the Jordan spin algebra L n (n 3). In the matrix algebras, the (canonical) inner product is given by X, Y = Re trace(xy ) and in L n, it is the usual inner product on R n. The symmetric cone of S n will be denoted by S n + (with a similar notation in other spaces). In any matrix algebra, let E ij be the matrix with ones in the (i, j) and (j, i) slots and zeros elsewhere; we write, E i := E ii. In a matrix algebra of rank r, {E 1, E 2,..., E r } is the canonical Jordan frame. We use the notation A = [a ij ] to say that A is a matrix with components a ij ; we also write A 0 A S r + and x 0 (> 0) x K (int(k)). For a given Jordan frame {e 1, e 2,..., e r } in V, we consider the corresponding Peirce decomposition ([5], Theorem IV.2.1) V = 1 i j r where V ii = R e i and V ij = {x V : x e i = 1 2 x = x e j}, when i j. For any x V, we get the corresponding Peirce decomposition x = i j x ij. We note that this is an orthogonal direct sum and x ii = x i e i for all i, where x i R. We recall a few new concepts and notations. Given a matrix A S r, a Euclidean Jordan algebra V and a Jordan frame {e 1, e 2,..., e r } with a specific ordering, we define the linear transformation D A : V V and Schur (or Hadamard) products in the following way: For any two elements x, y V with Peirce decompositions x = i j x ij and y = i j y ij, D A (x) = A x := V ij, 1 i j r a ij x ij V (4)

4 4 Taylor & Francis and I.T. Consultant and x y := r x ii, y ii E i x ij, y ij E ij S r. (5) 1 i<j r We immediately note that D A (x), x = A x, x = i j a ij x ij 2 = A, x x, (6) where the inner product on the far right is taken in S r. When V is S r with the canonical Jordan frame, the above products A x and x y coincide with the well-known Schur (or Hadamard) product of two symmetric matrices. This can be seen as follows. Consider matrices x = [α ij ] and y = [β ij ] in S r so that x = i j α ije ij and y = i j β ije ij are their corresponding Peirce decompositions with respect to the canonical Jordan frame. Then simple calculations show that A x = i j a ij α ij E ij = [a ij α ij ] and x y = [α ij β ij ], where the latter expression is obtained by noting that in S r, E ij, E ij = trace(e 2 ij ) = 2. It is interesting to note that (A x) x = A (x x), (7) where the bullet product on the right is taken in S r with respect to the canonical Jordan frame. Definition 2.1 A linear transformation L on V is said to be Peirce-diagonalizable if there exist a Jordan frame and a symmetric matrix A such that L = D A. Remarks (1) Note that in the above definition, both L and D A use a specific ordering of the Jordan frame. Any permutation of the elements of the Jordan frame results in a row and column permutation of the matrix A. (2) In the definition of D A above, we say that an entry a ij in A is relevant if V ij {0}. (When a ij is non-relevant, it plays no role in D A.) By replacing all nonrelevant entries of A by zero, we may rewrite D A = D B, where every non-relevant entry of B is zero. We also note that when V is simple, every V ij {0} (see [5], Corollary IV.2.4) and thus every entry in A is relevant. (3) Suppose V is a product of simple algebras. Then with appropriate grouping of the given Jordan frame, any element x in V can be decomposed into components so that the Peirce decomposition of x splits into Peirce decompositions of components in each of the simple algebras. Because of the uniqueness of the matrix B (of the previous remark), the transformation D B splits into component transformations of the same form, each of which is defined on a simple algebra. Any such component transformation corresponds to a block submatrix of a matrix obtained by permuting certain rows and columns of A. This type of decomposition will allow us, in certain instances, to restrict our analysis to transformations on simple algebras. (4) Let V be a simple algebra and Γ : V V be an algebra automorphism, that is, Γ is an invertible linear transformation on V with Γ(x y) = Γ(x) Γ(y) for all

5 Optimization Methods and Software 5 x, y V. Since such a Γ preserves the inner product in V ([5], p. 57), it is easily verified that Γ(x) Γ(y) = x y, (8) where Γ(x) Γ(y) is computed via the Peirce decompositions with respect to the Jordan frame {Γ(e 1 ), Γ(e 2 ),..., Γ(e r )}. Properties of the products A x and x y are described in [19] and [11], respectively. In particular, we have the following generalization of the classical Schur s theorem ([14], Theorem 5.2.1). Proposition 2.2 Suppose A 0 and x, y 0. Then It follows that A x 0 and x y 0. x = x ij 0 [ x ij 2 ] = x x 0. The following proposition is key to many of our results. Proposition 2.3 Suppose that V is simple. Then for any vector u R n +, there exists an x K such that x x = u u T. (9) Proof. We assume without loss of generality that r 2 and that the inner product is given by x, y = tr(x y) so that the norm of any primitive element is one. (This is because, in a simple algebra, the given inner product is a multiple of the trace inner product, see [5], Prop. III.4.1.) We produce an element y K whose Peirce decomposition with respect to {e 1, e 2,..., e r } is given by y = e 1 + e e r + i<j y ij with y ij = 2 for all i < j. (10) Then, for any u = (u 1, u 2,..., u r ) T R r +, we let x = r u i e i + ui u j y ij = P a (y), i<j 1 where a = r 1 ui e i. Since P a (K) K, we see that x K. A direct verification proves (9). Now to construct y, we proceed as follows. When V (which is simple) has rank 2, we take any y 12 V 12 with norm 2. Then the statement y = e 1 + e 2 + y 12 K follows from Lemma 1 in [12] or by the equality y 2 = 2y (the proof of which requires the use of the identity y12 2 = y (e 1 + e 2 ), see Prop. IV.1.4 in [5]). When the rank of V is more than 2, the (simple) algebra V is isomorphic to a matrix algebra. In this case, we may take the canonical Jordan frame as the given frame (see Remark 4 above) and let y ij = E ij for all i < j. Then y, which is the matrix of ones, is in K. This completes the proof of the proposition.

6 6 Taylor & Francis and I.T. Consultant Remark (5) In the construction of y above (when the rank is more than two), the use of the classification theorem of simple algebras can be avoided by the following argument (which is a slight modification of one) suggested by a referee. For 1 < i r, choose y 1i V 1i with y 1i = 2 and for 1 < i < j r, define y ij = 2y 1i y 1j. By Lemma IV.2.2 in [5], y ij = 2. Let y as in (10). We claim that y 2 = ry, proving the statement y K. To prove y 2 = ry, we induct on r. Write y = e 1 + v + w, where v = y y 1r, w = e 2 + e e r + 1<i<j r 1 y ij. Then w belongs to an appropriate simple algebra of rank r 1 and so w 2 = (r 1)w. Now using the standard properties of Peirce spaces (specifically, Prop. IV.1.4(i), Theorem IV.2.1(iii), Lemma IV.2.2 in [5]), we get e 2 1 = e 1, v 2 = (r 1)e 1 + w, 2e 1 v = v, 2v w = (r 1)v. This shows that y 2 = ry. By taking u to be the vector of ones in the above theorem, and using (7), we get the following. Corollary 2.4 When V is simple, for any A S r, there exists an x K such that A = (A x) x. 3. Some general properties From now on, unless otherwise specified, Peirce decompositions and D A are considered with respect to the Jordan frame {e 1, e 2,..., e r } in V. Our first result describes some general properties of D A. Theorem 3.1 For any A S r, the following statements hold: (1) D A is self-adjoint. (2) The spectrum of D A is σ(d A ) = {a ij : 1 i j r, a ij relevant}. (3) D A is diagonalizable. (4) If Γ is an algebra automorphism of V, then Γ 1 D A Γ is Peirce-diagonalizable with respect to the Jordan frame {Γ 1 (e 1 ), Γ 1 (e 2 ),..., Γ 1 (e r )} and with the same matrix A. (5) D A is a finite linear combination of quadratic representations corresponding to mutually orthogonal elements which have their spectral decompositions with respect to {e 1, e 2,..., e r }. (6) If A 0, then D A = k 1 P a i with 1 k r, where a i s are mutually orthogonal and have their spectral decompositions with respect to {e 1, e 2,..., e r }. In particular, if A is completely positive (that is, A is a finite sum of matrices of the form uu T, where u is a nonnegative vector), we may assume that a i K for all i. (7) If A 0, then D A (K) K; converse holds if V is simple. Proof. (1) This follows from the equality D A (x), y = i j a ij x ij, y ij = x, D A (y). (2) When V ij {0}, any nonzero x ij V ij acts as an eigenvector of D A with the corresponding eigenvalue a ij. Hence, all relevant a ij s form a subset of σ(d A ). Also, if x is an eigenvector of D A corresponding to an eigenvalue λ, by writing the Peirce decomposition of x with respect to {e 1, e 2,..., e r } and using the orthogonality of the Peirce spaces V ij, we deduce that λ must be equal to some relevant a ij. This proves that the spectrum of D A consists precisely of all relevant a ij s. (3) D A is a multiple of the Identity on each V ij. By choosing a basis in each V ij, we get a basis of V consisting of eigenvectors of D A. This proves that D A is diagonalizable.

7 Optimization Methods and Software 7 (4) Let Γ be an algebra automorphism of V ; let Γ 1 (e i ) = f i for all i, so that {f 1, f 2,..., f r } is a Jordan frame in V. Consider the Peirce decomposition y = i j y ij of any element y with respect to {f 1, f 2,..., f r }. Then Γ(y) = i j Γ(y ij) is the Peirce decomposition of Γ(y) with respect to {e 1, e 2,..., e r }. Hence, D A (Γ(y)) = i j a ijγ(y ij ). This implies that Γ 1 D A Γ(y) = i j a ij y ij. Thus, Γ 1 D A Γ is Peirce-diagonalizable with respect to the Jordan frame {f 1, f 2,..., f r } and with the matrix A. (5) Since A S r, we may write A as a linear combination of matrices of the form uu T, where u R r. From (2), (uu T ) x = i j u iu j x ij = P a (x), where a = r 1 u ie i. We see that D A is a linear combination of quadratic representations of the form P a, where a has its spectral decomposition with respect to {e 1, e 2,..., e r }. (6) When A 0, the expression for D A in Item (5) becomes a nonnegative linear combination. By absorbing the nonnegative scalars appropriately, we can write D A as a sum of quadratic representations. When A is completely positive, we may assume that u (which appears in the proof of Item (5)) is a nonnegative vector. In this case, the corresponding a belongs to K. (7) If A 0, then by Prop. 2.2, D A (K) K. For the converse, suppose V is simple. By Cor. 2.4, A = (A x) x for some x K. When D A (K) K, by Prop. 2.2, A x 0 and so (A x) x 0. Thus, A 0. Corollary 3.2 Suppose V is simple and for some a V, L a (K) K. Then a is a nonnegative multiple of the unit element. Proof. Using the spectral decomposition of a = r 1 a ie i, we may assume that L a = D A, where A = [ ai+aj 2 ]. Since L a (K) K, by the previous theorem, A is positive semidefinite. By considering the nonnegativity of any 2 2 principal minor of A, we conclude that all diagonal elements of A are equal. This gives the stated result. 4. The copositivity property In this section, we address the question of when D A is copositive on K. Recall that an n n real matrix A (symmetric or not) is a copositive (strictly copositive) matrix if x T Ax 0 (> 0) for all 0 x R n + and D A is copositive (strictly copositive) on K if D A (x), x 0 (> 0) for all 0 x K. We shall see that copositivity of D A is closely related to the cone of completely positive matrices. We first recall some definitions and introduce some notation. For any set Ω, let cone(ω) denote the convex cone generated by Ω (so that cone(ω) is the set of all finite nonnegative linear combinations of elements of Ω). Let COP n := The set of all copositive matrices in S n, CP n := cone{uu T : u R n +}, N N n := The set of all nonnegative matrices in S n, DN N n := S n + N N n, and D(K) := cone{x x : x K}.

8 8 Taylor & Francis and I.T. Consultant We note that CP n is the cone of completely positive matrices and DN N n is the cone of doubly nonnegative matrices; for information on such matrices, see [1]. Remark (6) When V is simple, D(K) is independent of the Jordan frame used to define x x. This follows from (8). It is also independent of the underlying inner product in V. A similar statement can be made for general algebras (as they are products of simple ones). It is well-known, see [1], p. 71, that the cones COP n, CP n, and DN N n are closed. We prove a similar result for D(K). Proposition 4.1 Let r denote the rank of V and N = r(r+1) 2. Then (i) D(K) = { N 1 x i x i : x i K, 1 i N}, (ii) D(K) is closed in S r, and (iii) D(K) DN N r. (iv) When V is simple, CP r D(K) DN N r with equality for r 4. Proof. (i) This follows from a standard application of Carathéodory s Theorem for cones, see [1], p. 46. (ii) To show that D(K) is closed, we take a sequence {z k } D(K) S r that converges to z S n. Then, with respect to the canonical Jordan frame in S r, we have (z k ) ij z ij for all i j. Now, z k = N m and k. Then m=1 x(k) m x (k) m with x (k) m K for all (z k ) ij = N ( m=1 x (k) m x (k) m )ij N = ε ij (x m (k) ) ij 2 E ij, m=1 and ε ij is a positive num- converges for each m and where i j (x(k) m ) ij is the Peirce decomposition of x (k) m ber that depends only on the pair (i, j). As (z k ) ij (i, j), (x (k) m ) ij is bounded as k. We may assume, as V ij s are closed, that (x m (k) ) ij converges to, say, (x m ) ij in V ij. Defining x m := i j (x m) ij, we see that x (k) m x m. Then x m K and x (k) m x (k) m m=1 x m x m D(K). Thus D(K) is closed. x m x m for each m. We conclude that z = N (iii) Every matrix in D(K) is positive semidefinite (from Prop. 2.2) and has nonnegative entries (from the definition of x x). Thus we have the stated inclusion. (iv) When V is simple, the inclusion CP r D(K) follows from Prop When r 4, it is known, see [1], p. 73, that CP r = DN N r. Remarks (7) For r 5, we have the inclusions CP r D(S r +) D(H r +) D(Q r +) DN N r. While CP r DN N r for r 5, see the example given below, it is not clear if the other inclusions are indeed proper. In particular, it is not clear if x x is completely positive for every x S r +. (8) Consider the following matrices:

9 Optimization Methods and Software M = and N = It is known that M and N are doubly nonnegative; while N completely positive, M not completely positive, see [1], p. 63 and p. 79. It can be easily shown that M and N are not of the form x x for any x S 5 +. (If M = x x for some x S 5 +, then the entries of x are the square roots of entries of M. The leading 3 3 principal minor of any such x is negative.) It may be interesting to see if M belongs to D(S 5 +) or not. Theorem 4.2 Suppose V is simple and D A is copositive on K. Then A is a copositive matrix. Proof. Since V is simple, we can apply Prop. 2.3: For any u R r +, there is an x K such that x x = u u T. Then by (6), 0 D A (x), x = A, x x = A, uu T = u T Au. Thus, A is a copositive matrix. Remark (9). It is clear that the converse of the above result depends on whether the equality CP r = D(K) holds or not. 5. The Z-property A linear transformation L on V is said to be a Z-transformation if x 0, y 0, and x, y = 0 L(x), y 0. Being a generalization of a Z-matrix, such a transformation has numerous eigenvalue and complementarity properties, see [10] and [12]. In this section, we characterize the Z-property of a Peirce-diagonalizable transformation. Corresponding to a Jordan frame {e 1, e 2,..., e r }, we define two sets in V : and C := {x y : x 0, y 0}, Ĉ := {x y : 0 x y 0}. Now, define the linear transformation Λ : S r S r by Λ(Z) := EZE = δ(z) E, where Z = [z ij ], δ(z) = i,j z ij, and E denotes the matrix of ones. It is clear that Λ is self-adjoint on S r. Lemma 5.1 When V is simple with rank r, we have C = S r + and Ĉ = Sr + Ker(Λ). Proof. From Prop. 2.2, C S r +. To see the reverse inclusion, let A S r +. By Cor. 2.4, there is an x K such that A = (A x) x. By Prop. 2.2, y := A x 0 and so A = y x, where x, y K. Thus, S r + C and the required equality holds.

10 10 Taylor & Francis and I.T. Consultant Now, let B = x y Ĉ. Then δ(b) = i,j b ij = i j x ij, y ij = x, y = 0 and so B Ker(Λ). Since B C = S+, r we have B S+ r Ker(Λ). For the reverse inclusion, let B S+ r Ker(Λ). As C = S+, r we can write B = x y, where x, y K. Then x, y = δ(b) = 0 and so B Ĉ. This completes the proof. Theorem 5.2 Let V be simple and A S r. Then D A has the Z-property if and only if there exist B k S r + and a sequence α k R for which A = lim k (α k E B k ). In this case, D A = lim k (α k I D Bk ). Proof. From the previous lemma, Ĉ = Sr + Ker(Λ). Then the dual of Ĉ is given by, see e.g, [1], p. 48, (Ĉ) = S r + + Ran(Λ), where the overline denotes the closure. Now, D A has the Z-property if and only if 0 x y 0 D A (x), y 0. Writing D A (x), y = i j a ij x ij, y ij = trace(az), where Z = x y Ĉ, we see that D A has the Z-property if and only if Z Ĉ A, Z 0. This means that D A has the Z-property if and only if A (Ĉ) = S+ r + Ran(Λ). The stated conclusions follow. Remark (10) It is known, see [18], that any Z-transformation is of the form lim k (α k I S k ), where the linear transformation S k keeps the cone K invariant. Our theorem above reinforces this statement with an additional information that S k is once again Peirce-diagonalizable. 6. Complementarity properties In this section, we present some complementarity properties of D A. Theorem 6.1 For an A S r, consider the following statements. (i) D A is strictly (=strongly) monotone on V : D A (x), x > 0 for all x 0. (ii) D A has the GUS-property: For all q V, LCP(D A, K, q) has a unique solution. (iii) D A has the P-property: For any x, if x and D A (x) operator commute with x D A (x) 0, then x = 0. (iv) Every relevant entry of A is positive. Then (i) (ii) (iii) (iv). When V is simple, the reverse implications hold. Proof. The implications (i) (ii) (iii) are well-known and true for any linear transformation, see [9], Theorem 11. Now suppose (iii) holds. Since D A is self-adjoint and satisfies the P-property, all its eigenvalues are positive in view of Theorem 11 in [9]. Item (iv) now follows from Theorem 3.1, Item (2). Now assume that V is simple. Then every entry of A is

11 Optimization Methods and Software 11 relevant and so when (iv) holds, we have D A (x), x = a ij x ij 2 > 0 x 0. i j This proves that (iv) (i). Remark (11) Suppose that V is simple and a V. By writing the spectral decomposition a = a 1 e 2 + a 2 e a r e r, we consider L a and P a with corresponding matrices [ ai+aj 2 ] and [a i a j ]. Applying the above result, we get two known results: L a has the P-property if and only if a > 0 and P a has the P-property if and only if ±a > 0. In what follows, we use the notation A R 0 to mean that A has the R 0 -property, that is, LCP(A, R n +, 0) has a unique solution (namely zero). A similar notation is used for D A in relation to LCP(D A, K, 0). Theorem 6.2 Suppose A 0. If A R 0, then D A R 0. Converse holds when V is simple. Proof. Assume that A R 0, but D A R 0. Then there is a nonzero x V such that x 0, D A (x) 0, and D A (x), x = 0. We have from (6), 0 = D A (x), x = A, x x. As x 0 in V, X := x x 0 (by Prop. 2.2). Now in S r, X 0, A 0, and A, X = 0 AX = 0. Since X, which is nonzero, consists of nonnegative entries, for any nonzero column u in X, we have Au = 0. This u is a nonzero solution of the problem LCP(A, R r +, 0), yielding a contradiction. Hence, D A R 0. For the converse, suppose that V is simple and D A R 0. Assume, if possible, that A R 0. Then there exists a nonzero u R r such that u 0, Au 0, and Au, u = 0. As V is simple, for this u, by Prop. 2.3, there is an x K such that x x = uu T. Since u is nonzero, x is also nonzero. Now D A (x), x = A, x x = A, uu T = u T Au = 0. We also have D A (x) = A x 0. Thus, even in this case, D A R 0. Hence, when V is simple, D A R 0 A R 0. Remark (12) Theorem 6.2 may not hold if A is not positive semidefinite. For example, consider the matrices A = [ ] and C = [ ] with the corresponding transformations D A (x) = A x and D C (x) = C x on S 2. It is easily seen that A is in R 0 while D A R 0 and D C is in R 0 while C R 0.

12 12 Taylor & Francis and I.T. Consultant Before formulating our next result, we recall some definitions. Given a linear transformation L on V, its principal subtransformations are obtained in the following way: Take any nonzero idempotent c in V. Then the transformation T c := P c L : V (c, 1) V (c, 1) is called a principal subtransformation of L, where V (c, 1) = {x V : x c = x}. (We note here that V (c, 1) is a subalgebra of V.) We say that L has the completely Q-property if for every nonzero idempotent c, the subtransformation T c has the Q-property on V (c, 1). The transformation L is said to have the S-property if there exists a d > 0 in V such that L(d) > 0. The completely S-property is defined by requiring that all principal subtransformations have the S-property. It is well known, see e.g., [6], that for a symmetric positive semidefinite matrix A, the R 0 and Q properties are equivalent to the strict copositivity property on R n. A related result, given by Malik [17] for linear transformations on Euclidean Jordan algebras, says that for a self-adjoint, cone-invariant transformation, R 0, completely Q, and completely S properties are equivalent to strict copositivity property. These two results yield the following Theorem 6.3 With A 0, consider the following statements: (a) A has the Q-property. (b) A is strictly copositive. (c) A has the R 0 -property. (d) D A has the R 0 -property. (e) D A has the completely S-property. (f) D A has the completely Q-property. (g) D A is strictly copositive. (h) D A has the Q-property. Then (a) (b) (c) (d) (e) (f) (g) (h). Proof. As A is symmetric and positive semidefinite, the equivalence of (a), (b), and (c) is given in [6]; the implication (c) (d) is given in the previous result. As A is positive semidefinite, D A (which is self-adjoint) keeps the cone invariant. The equivalence of (d) (g) now follows from Malik [17], Theorem and Corollary Finally, the implication (g) (h) follows from the well-known theorem of Karamardian, see [15]. Remark (13) One may ask if the reverse implications hold in the above theorem. Since D A involves only the relevant entries of A, easy examples (e.g., V = R 2 = S 1 S 1 ) can be constructed in the non-simple case to show that (h) need not imply (a). While we do not have an answer to this question in the simple algebra case, in the result below we state some necessary conditions for D A to have the Q-property. Proposition 6.4 Suppose D A has the Q-property. Then the following statements hold: (i) The diagonal elements of A are positive. (ii) When V is simple, no positive linear combination of two rows of A can be zero. (iii) If V has rank 2 and A 0, then D A has the R 0 -property. Proof. (i) Since D A has the Q-property, it has the S-property, that is, there is a u > 0 in V such that D A (u) > 0. Writing the Peirce decompositions u = i j u ij

13 Optimization Methods and Software 13 and D A (u) = i j a iju ij, we see that u i > 0 and a ii u i > 0 for all i, where u ii = u i e i. Thus, the diagonal of A is positive. Now, let b := r 1 (a 1 ii ) 1 4 e i and consider D A = P b D A P b. It is easily seen that D A has the Q-property and Peircediagonalizable with the matrix whose entries are aiia ij a jj. Note that the diagonal elements of this latter matrix are one. (ii) Suppose V is simple and let, without loss of generality, λa 1 + µa 2 = 0, where λ and µ are positive numbers, and A 1, A 2, respectively, denote the first and second rows of A. Let a = λe 1 + µe 2 +e 3 +e 4 + +e r and consider D A := P a D A P a. Then D A has the Q-property and is Peirce-diagonalizable with a matrix in which the sum of the first two rows is zero. Thus, we may assume without loss of generality that in the given matrix A, the sum of first two rows is zero. As V is simple, the space V 12 is nonzero. Let q 12 be any nonzero element in V 12. We claim that LCP(D A, K, q 12 ) does not have a solution. Assuming the contrary, let x be a solution to this problem and let y = D A (x) + q 12. Then x 0, y 0, and x y = 0. Writing the Peirce decompositions of x and y and using the properties of the spaces V ij (see [5], Theorem IV.2.1), we see that 0 = (x y) 12 = 1 2 (a 11 + a 12 )x 1 x (a 12 + a 22 )x 2 x 12 + (a 1i + a 2i ) x 1i x 2i (x 1 + x 2 )q 12, 2<i where x 11 = x 1 e 1 and x 22 = x 2 e 2. Since the sum of first two rows of A is zero, the above expression reduces to 0 = 1 2 (x 1 + x 2 )q 12. As q 12 is nonzero, we get x 1 + x 2 = 0. Now, since x 0, we must have x 1 0 and x 2 0. Thus, we have x 1 = 0 = x 2. This leads, from x 0, to x 1j = 0 = x 2j for all j, see [7], Prop But then y 0 together with y 11 = a 11 x 11 = 0 leads to y 1j = 0 for all j, and, in particular, to y 12 = 0. Now we see that 0 = y 12 = a 12 x 12 + q 12 = q 12, which contradicts our assumption that q 12 is nonzero. Hence, D A does not have the Q-property, contrary to our assumption. This proves the stated assertion. (iii) Now assume that V has rank 2. In this case, A is a 2 2 matrix. Without loss of generality (see the proof of Item (i)), assume that the diagonal of A consists of ones. Given that D A has the Q-property, we claim that A has the R 0 -property and (via the previous theorem) D A has the R 0 -property. Assume, if possible, that there is a nonzero vector d R 2 such that d 0, Ad 0, and d, Ad = 0. As A is symmetric and positive semidefinite, these conditions lead to Ad = 0. Since each diagonal element of A is one, d cannot have just one nonzero component. Thus, both components of d are nonzero. In this case, a positive linear combination of rows of A would be zero. This implies, see Item (ii), that V must be non-simple. In this case, V is isomorphic to R 2 and we may write for any x, x = x 1 e 1 + x 2 e 2, D A (x) = a 11 x 1 e 1 + a 22 x 2 e 2. Thus, we may regard D A as a diagonal matrix acting on R 2. Now, it is easy to show (from the Q-property) that D A has the R 0 -property. This completes the proof. In the next two remarks, we show how to recover known results from the above proposition. Remarks (14) Suppose for some a V, D A = L a has the Q-property. By writing the spectral decomposition a = a 1 e 1 + a 2 e a r e r, we may let A = [ ai+aj 2 ]. By Item (i) of the above proposition, a i > 0 for all i, that is, a > 0. Then, L a (x), x = a, x 2 > 0 for all x 0. Thus, L a is strictly monotone.

14 14 REFERENCES (15) Suppose, for some a V, D A = P a has the Q-property. Suppose further that V is simple. By writing the spectral decomposition a = a 1 e 1 + a 2 e a r e r we may let A = [a i a j ]. Then a i 0 for all i, by Item (i). If some a i and a j have opposite signs, then a positive linear combination of the rows of A corresponding to i and j would be zero, contradicting Item (ii). Thus all a i have the same sign. This means that ±a > 0 and A = [a i a j ] is a matrix with positive entries; consequently, by Theorem 6.1, D A is strictly monotone. Now suppose that V is a product of simple algebras V i, i = 1, 2..., m. Then we can write P a as a product of transformations P a (i), where a (i) V i for all i. Since P a has the Q-property, it follows that every P a (i) has the Q-property on V i. By our earlier argument P a (i) is strictly monotone on V i and so P a will also be strictly monotone on V. Acknowledgments We thank the referees for their constructive comments which improved and simplified the presentation of the paper. References [1] A. Berman and Noami Shaked-Monderer, Completely Positive Matrices, World Scientific, New Jersey, [2] C.B. Chua and P. Yi, A continuation method for nonlinear complementarity problems over symmetric cones, SIAM J. Optim. 20 (2010) pp [3] R.W. Cottle, J.-S. Pang, and R. Stone, The Linear Complementarity Problem, Academic Press, Boston, [4] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, [5] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Clarendon Press, Oxford, [6] M.S. Gowda, On Q-matrices, Math. Prog. 49 (1990), pp [7] M.S. Gowda and R. Sznajder, Automorphism invariance of P and GUS-properties of linear transformations on Euclidean Jordan algebras, Math. Oper. Res. 31 (2006), pp [8] M.S. Gowda and R. Sznajder, Some global uniqueness and solvability results for linear complementarity problems over symmetric cones, SIAM J. Optim. 18 (2007), pp [9] M.S. Gowda, R. Sznajder, and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear Alg. Appl. 393 (2004), pp [10] M.S. Gowda and J. Tao, Z-transformations on proper and symmetric cones, Math. Prog. Ser. B 117 (2009), pp [11] M.S. Gowda and J. Tao, Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras, preprint (2010), to appear in Positivity. Available at gowda. [12] M.S. Gowda, J. Tao, and G. Ravindran, Some complementarity properties of Z- and Lyapunov-like transformations on symmetric cones, Research Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland, USA, February 2010 (Revised February 2011). Available at gowda. [13] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, [14] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [15] S. Karamardian, An existence theorem for the complementarity problem, J. Optim. theory Appl. 19 (1976), pp [16] A. Korányi, Monotone functions on formally real Jordan algebras, Math. Ann., 269 (1984), pp [17] M. Malik, Some geometrical aspects of the cone linear complementarity problem, PhD Thesis, Indian Statistical Institute, New Delhi, India, [18] H. Schneider and M. Vidyasagar, Cross-positive matrices, SIAM J. Numer. Anal., 7 (1970), pp [19] R. Sznajder, M.S. Gowda, and M.M. Moldovan, More results on Schur complements in Euclidean Jordan algebras, preprint (2009), to appear in Jour. Global Optim. Available at gowda. [20] Y. Zhou and M.S. Gowda, On the finiteness of the cone spectrum of certain linear transformations on Euclidean Jordan algebras, Linear Alg. Appl., 431 (2009), pp