M. Seetharama Gowda. J. Tao. G. Ravindran

Size: px
Start display at page:

Download "M. Seetharama Gowda. J. Tao. G. Ravindran"

Transcription

1 Some complementarity properties of Z and Lyapunov-like transformations on symmetric cones M. Seetharama Gowda Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 2250, USA gowda@math.umbc.edu J. Tao Department of Mathematical Sciences, Loyola University Maryland, Baltimore, MD 220, USA jtao@loyola.edu G. Ravindran Indian Statistical Institute, Chennai, India gomatamr@hotmail.com This article deals with some complementarity properties of Z and Lyapunov-like transformations on a symmetric cone. Similar to the results proved for Lyapunov and Stein transformations on Herm(R n n ), we show that for Lyapunov-like transformations, the P and S properties are equivalent, and that for a Z- transformation, the S-property implies the P-property of a transformation that is (quadratically) similar to the given transformation. In addition, we show that a Lyapunov-like transformation is the sum of a Lyapunov transformation and a derivation. Key words: Euclidean Jordan algebra; symmetric cone; complementarity problems; Z and Lyapunov-like transformations; P and GUS properties MSC2000 Subject Classification: Primary: 90C33, 7C55; Secondary: 5A48 OR/MS subject classification: Primary: Programming; Secondary: Mathematics. Introduction Given a closed convex set K in a (finite dimensional) real Hilbert space (H,, ), a function f : K H, and an element q H, the variational inequality problem VI(f, K, q) is to find an x H such that x K and f(x ) + q, x x 0 for all x K. (.) This problem has been well studied in the literature and appears in numerous applications, see [6], [9], [8]. When K is a closed convex cone, the above problem reduces to a cone complementarity problem: Find x H such that x K, f(x ) + q K, and x, f(x ) + q = 0, (.2) where K := {x H : x, y 0, y K} is the dual of K. This further reduces to the so-called cone linear complementarity problem (cone LCP) when f is linear and to the nonlinear complementarity problem (NCP) when H = R n with the usual inner product and K = R+. n Two important cone LCPs are the standard LCP (this is NCP with f linear) and semidefinite LCP (when H is the space of all n n real symmetric matrices and K is the cone of positive semidefinite matrices in H). One of the most basic and longstanding problems in this area is to characterize (or describe conditions on) f and K so that VI(f, K, q) has a unique solution for all q. Three well known uniqueness results/conditions are the following: () f is strongly monotone and K is arbitrary (Theorem 2.3.3, [6]). (2) In the setting of standard LCP, f(x) = Mx, where M is a P-matrix, that is, all principal minors of M are positive, or equivalently, x Mx 0 x = 0 [8], [0]. (3) K is polyhedral, f is linear and coherently oriented [20]. 0

2 Gowda, Tao, Ravindran: Complementarity properties Going beyond polyhedral sets/cones, in recent times, some effort has been made to understand this uniqueness problem over symmetric cones in Euclidean Jordan algebras, in particular over the semidefinite cone and the Lorentz (or second order) cone. Item (2) above has the following analogue in the symmetric cone setting [4]: Let V be a Euclidean Jordan algebra with the corresponding symmetric cone K, L : V V be linear. Then VI(L, K, q), which is written as LCP(L, K, q), has a unique solution for all q V in which case, we say that L has the globally uniquely solvable (GUS) property if and only if L has the P-property, that is, [x and L(x) operator commute, and x L(x) 0 ] x = 0; L has the cross-commutative property, that is, for any q and any two solutions x and x 2 of LCP(L, K, q), x operator commutes with L(x 2 ) + q and x 2 operator commutes with L(x ) + q. Although the above cross-commutative property is obvious in the case of the Euclidean Jordan algebra R n and for monotone L [4], it is not easily verifiable in the general case. A uniqueness result avoiding the cross-commutative property is yet to be obtained and, at present, the work has been limited to understanding the P-property for certain classes of linear transformations on symmetric cones. The present paper continues this line of work; we investigate the P-property for the classes of Z and Lyapunov-like transformations. A linear transformation L : V V is said to be a Z-transformation (or said to have the Z-property) if x 0, y 0, and x, y = 0 L(x), y 0, (.3) and Lyapunov-like if both L and L are Z-transformations. The Z-transformations, the negative of which are called cross-positive transformations in [22], are generalizations of Z-matrices and have numerous complementarity properties, see the recent paper [5]. In particular, it was shown in [5] that for a Z-transformation L, the following are equivalent, and moreover, they hold when, additionally, L has the P-property: (a) L has the Q-property, that is, for all q V, LCP(L, K, q) has a solution. (b) L has the S-property, that is, there exists d > 0 such that L(d) > 0. (c) L has the positive stable property, that is, all eigenvalues of L have positive real parts. (d) L is invertible with L (K) K. This paper is concerned with the converse issue: Whether these conditions imply the P-property for Z-transformations. Our motivation comes from the following particular cases: (i) On the algebra R n, a Z-matrix has the P-property if and only if it has the S-property, see Theorem 3..0, [2]. (ii) On the Jordan spin algebra L n, a Z-transformation has the P-property if and only if it has the S-property, see Theorem 3, [5]. (iii) On the algebra Herm(R n n ) of all n n real symmetric matrices, the Lyapunov transformation L A, defined by L A (X) = AX + XA T (for a given n n real matrix A), has the P-property if and only if it has the S-property [2]. (We note that L A has the Lyapunov-like property and the S-property is related to the asymptotic stability of the continuous linear dynamical system ẋ + Ax = 0.) (iv) On the algebra Herm(R n n ), the Stein transformation S A, defined by S A (X) = X AXA T (for a given n n real matrix A), has the P-property if and only if it has the S-property []. (We note that S A has the Z-property and the S-property is related to the asymptotic stability of the discrete linear dynamical system x(k + ) = Ax(k), k =, 2,...)

3 2 Gowda, Tao, Ravindran: Complementarity properties (v) For a Lyapunov-like transformation L on a Euclidean Jordan algebra, L has the GUSproperty if and only if L is positive stable and positive semidefinite, see Theorem 7., [3]. In this paper, we show that for Lyapunov-like transformations, the P and S properties are equivalent, and that for a Z-transformation, the S-property implies the P-property of transformation (quadratically) similar to the given transformation. In [3], Damm has shown that on Herm(R n n ) (Herm(C n n )), a Lyapunov-like transformation is of the form L A for some A R n n (respectively, A C n n ). Based on Lie algebraic ideas, we show that Lyapunov-like transformations on a general Euclidean Jordan algebra are of the form L = L a + D, where a V and D is a derivation on V. This will allow us to extend Damm s result to Herm(H n n ) (the algebra of all n n Hermitian quaternionic matrices) and describe Lyapunov-like transformations on the Jordan spin algebra L n. Here is an outline of our paper. In Section 2, we briefly recall concepts and notations from the Euclidean Jordan algebra theory. Section 3 deals with the P-property of a Z-transformation. In Section 4, we characterize Lyapunov-like transformations and describe them on various simple Euclidean Jordan algebras. Finally Section 5 deals with the characterization of the P-property of a Lyapunov-like transformation. 2. Euclidean Jordan Algebras Throughout this paper, we let (V,,, ) denote a Euclidean Jordan algebra of rank r. Recall that V is a finite dimensional real Hilbert space with the bilinear Jordan product - written x y - satisfying the following properties for all x, y, z V : (i) x y = y x, (ii) x (x 2 y) = x 2 (x y), and (iii) x y, z = y, x z. We refer to Faraut and Korányi [7], Gowda, Sznajder, and Tao [4], Schmieta and Alizadeh [9] for basic definitions, concepts, and results. In V we write x y if x, y = 0. The symmetric cone of V is the (closed convex self-dual) cone of squares K := {x 2 : x V }. We use the notation x 0 (x > 0) when x K (respectively, x interior(k)). As is well known, any Euclidean Jordan algebra is a product of simple Euclidean Jordan algebras and every simple algebra is isomorphic to one of the following: () The Jordan spin algebra L n ; (2) The algebra Herm(R n n ) of n n real symmetric matrices; (3) The algebra Herm(C n n ) of all n n complex Hermitian matrices; (4) The algebra Herm(H n n ) of all n n quaternion Hermitian matrices; (5) The algebra Herm(O 3 3 ) of all 3 3 octonion Hermitian matrices. Here, the symbols R, C, H, and O, denote, respectively, the set of all real numbers, complex numbers, quaternions, and octonions. In all of the above matrix algebras, the inner product is given by X, Y = Re tr (XY ) and the Jordan product is given by X Y = 2 (XY + Y X). Recall that a set {e, e 2,...,e r } is a Jordan frame in V when each e i is a primitive idempotent, e i e j = δ ij, and r e i = e, where e is the identity element in V. Given any a V, we let L a and P a denote the corresponding Lyapunov transformation and quadratic representation of a on V : L a (x) := a x and P a (x) := 2a (a x) a 2 x. These are linear and self-adjoint on V. In addition, P a (K) K for any a. We say that elements a and b in V operator commute if L a L b = L b L a. It is known that a and b operator commute if and only if they have their spectral decompositions with respect to a common Jordan frame. We recall the spectral and Peirce decompositions of an element in V.

4 Gowda, Tao, Ravindran: Complementarity properties 3 Theorem 2. (Spectral decomposition; Theorem III..2, [7]) Let V be a Euclidean Jordan algebra with rank r. Then for every x V, there exists a Jordan frame {e,..., e r } and real numbers λ,...,λ r such that x = λ e + + λ r e r. (2.) Here, the numbers λ i are the eigenvalues of x and the expression λ e + + λ r e r is the spectral decomposition (or the spectral expansion) of x. Given (2.), the trace of x is defined by tr(x) = λ + + λ r. It is well known that x, y tr := tr(x y) defines another inner product compatible with the Jordan product. Also, when V is simple, there exists a positive number α such that, x, y = α tr(x y) for all x and y (see, Prop. III.4., [7]). More generally, if V is a product of simple algebras V i (i =, 2,...,N) (where the Jordan product is computed componentwise and the inner product is the sum of the inner products in each V i ), then for two objects x = (x, x 2,...,x N ) and y = (y, y 2,..., y N ) in V, N N N x, y = x i, y i = α i tr(x i y i ) = α i x i, y i tr. (2.2) We also note that x, y tr = N x i, y i tr. (2.3) It follows from these that x 0, y 0, x, y = 0 x i 0, y i 0, and x i, y i = 0 i. Let {e, e 2,...,e r } be a fixed Jordan frame in V. For i, j {, 2,...,r}, consider the eigenspaces V ii := {x V : x e i = x} = Re i and when i j, Then we have the following V ij := {x V : x e i = 2 x = x e j}. Theorem 2.2 (Peirce decomposition; Theorem IV.2., [7]) The space V is the orthogonal direct sum of spaces V ij (i j). Furthermore, V ij V ij V ii + V jj, V ij V jk V ik if i k, and V ij V kl = {0} if {i, j} {k, l} =. Thus, given any Jordan frame {e, e 2,..., e r }, we can write any element x V as x = x i e i + i= i<j where x i R and x ij V ij. We refer to this as the Peirce decomposition of x with respect to the given Jordan frame. We will also use the following notation: For k r, V (e + e e k, ) := V ij. x ij i j k From now on, we use a boldface letter to denote either a class of linear transformations or a property satisfied by a linear transformation; for example, L P if and only if L has the P-property.

5 4 Gowda, Tao, Ravindran: Complementarity properties 3. Z-transformations and the P-property Recall that a linear transformation on a Euclidean Jordan algebra V is a Z-transformation if it satisfies condition (.3). This condition can also be described in other equivalent ways: Theorem 3. The following are equivalent: (i) L has the Z-property. (ii) For every Jordan frame {e, e 2,..., e r } in V, it holds that L(e i ), e j 0 for all i j. (iii) e tl (K) K for all t 0 in R. (iv) Any trajectory of the differential equation ẋ + L(x) = 0 which starts in K stays in K. (v) There exists α n in R and linear transformations S n on V such that S n (K) K and L = lim n (α ni S n ). In the above result, the equivalence of (i) and (ii) is given in [5] and the equivalence of (i), (iii), and (iv) can be found in [22], [5]. Our primary example of a Z-transformation is L = I S, where S is a linear transformation satisfying S(K) K. If L and L are both Z-transformations, then L is said to be Lyapunov-like; an example is the Lyapunov transformation L A on Herm(R n n ). In Section 4, we characterize such transformations. The result below shows that the Z-property is independent of the inner product. Theorem 3.2 Suppose L has the Z-property on the Euclidean Jordan algebra V which carries the inner product,. Then L has the Z-property with respect to the canonical inner product, tr and conversely. Proof. The result is obvious when V is simple, as any inner product on a simple algebra is a positive multiple of the trace inner product. We assume that V is a product of simple algebras; for simplicity, let V = V V 2, where V and V 2 are simple. Assume that L has the Z-property with respect to, and consider x = (x, x 2 ) 0, y = (y, y 2 ) 0 with 0 = x, y tr = x, y tr + x 2, y 2 tr. This implies that x i, y i tr = 0 for i =, 2 and hence (via (2.2)), 0 x (y, 0) 0 with respect to,. Hence L(x), (y, 0) 0, which yields L(x), y tr 0. A similar inequality ensues when we replace the subscript by 2. By adding these inequalities and using (2.3), we get L(x), y tr 0. This proves that L has the Z-property with respect to the trace inner product. The converse statement is proved in a similar way. As mentioned in the Introduction, Z P Z S. The reverse inclusion holds in the algebras R n and L n, and for any Lyapunov transformation L A on Herm(R n n ). We address the general situation here. In the result below, e denotes the unit element in V and V carries the trace inner product, in which case, the norm of any primitive idempotent is one. Theorem 3.3 If L Z and L(e) > 0, then L P. More generally, if L Z S, then there is a c > 0 such that P c LP c P. Proof. First suppose that L Z and L(e) > 0. Let x V be such that x and L(x) operator commute with x L(x) 0.

6 Gowda, Tao, Ravindran: Complementarity properties 5 Then for some Jordan frame {e, e 2,..., e r } in V we have the spectral decompositions, x = x i e i and L(x) = y i e i, where x i y i 0 for all i =, 2,...,r. Hence, x i L(e i ) = L(x) = which leads to y i e i, x i L(e i ), e j = y j j =, 2,..., r, (3.) as e i s are orthogonal and e j = for all j. Now, in view of Theorem 3.(ii), the matrix A := [a ij ] with a ij = L(e i ), e j is a Z-matrix and A T p = q, where p (q) is the column vector in R r whose components are x i (respectively, y i ). As x i y i 0 for all i =, 2,...,r, we see that p A T p 0, where denotes the Hadamard (or the componentwise) product. Now the assumption L(e) > 0 implies that L(e i ), e j = L(e), e j > 0 j =, 2,...,r. This leads to i= A T > 0 (in R r ), where is the vector in R r with all components equal to one. This means that the Z-matrix A T is also an S-matrix. It follows that, see Theorem 3..0 in [2], A T is a P-matrix and hence p A T p 0 p = 0; This proves that x i = 0 for all i. Hence L has the P-property. Now for the general case, assume that L Z and that there is a d > 0 such that L(d) > 0. Put c := d > 0 and consider the quadratic representation P c. Then P c is self-adjoint, invertible, (P c ) = P c, P c (K) = K, and P c (e) = c 2 = d. It is easily seen that L := Pc LP c is a Z-transformation with L(e) = (P c ) (L(d)) > 0. By the first part of the result, L has the P-property. This completes the proof. Remark. It is not known if in the above result, Pc LP c P can be replaced by L P proving the result Z S = Z P. 4. Lyapunov-like transformations Recall that a linear transformation L on V is Lyapunovlike if both L and L are Z-transformations, i.e., if 0 x y 0 L(x), y = 0. In view of Theorem 3., this condition is equivalent to: or For any Jordan frame {e, e 2,..., e r }, L(e i ), e j = 0 i j e tl (K) K t R. Note that the latter condition can be written as e tl (K) = K for all t R, or e tl Aut(K) t R, (4.) where Aut(K) is the set of all (invertible) linear transformations which map K onto K. Since Aut(K) can be regarded as a matrix group, the above condition immediately says (see Section 7.6 in []) that

7 6 Gowda, Tao, Ravindran: Complementarity properties L is a Lyapunov-like transformation if and only if it belongs to the Lie algebra of Aut(K). As we shall see below, the above result allows us to characterize Lyapunov-like transformations. We begin with our first result that a Lyapunov-like transformation on a product algebra is a product of Lyapunov-like transformations. Theorem 4. Suppose that V is the product of algebras V k, k =, 2,..., N, and L : V V is Lyapunov-like. Then, identifying V k with {0} V k {0}, we have L(V k ) V k and L restricted to any V k is Lyapunov-like. Proof. For simplicity, let V = V V 2. Consider any 0 x V. Then for any 0 y 2 V 2, 0 (x, 0) (0, y 2 ) 0 in V. By the Lyapunov-like property, L(x, 0), (0, y 2 ) = 0. As y 2 is arbitrary, it follows that the second component of L(x, 0) is zero. This proves that L(x, 0) V {0}. By linearity and our identification, L(V ) V. Similarly, L(V 2 ) V 2. The Lyapunov-like property of L on each V k is easily verified. Remark. If, in the above theorem, we let L k denote the restriction of L to V k, then L(x, x 2,...,x N ) = (L (x ), L 2 (x 2 ),..., L N (x N )), so we may regard the Lyapunov-like transformation L as a product of Lyapunov-like transformations L k. Conversely, if each L k is Lyapunov-like on V k, then L defined above is Lyapunov-like on V. We now relate Lyapunov-like transformations with another class of transformations known as derivations. A linear transformation D on V is said to be a derivation if for all x, y V, D(x y) = D(x) y + x D(y). It is well-known, see Page 36, [7], that D is a derivation on V if and only if e td Aut(V ) t R, where Aut(V ) is the group of all algebra automorphisms of V (these are invertible linear transformations on V that preserve the Jordan product). This means that Der(V ), the set of all derivations, is the Lie algebra corresponding to the group Aut(V ). Recalling that for an element a V, the corresponding Lyapunov transformation L a is defined by L a (x) = a x, any finite linear combination of commutators of the form [L a, L b ] := L a L b L b L a is a derivation, called the inner derivation. It is known that on a Euclidean Jordan algebra, every derivation is inner, see, Prop. VI..2, [7] or Theorem 8, [6]. Here is our characterization result. Theorem 4.2 A linear transformation L on a Euclidean Jordan algebra V is Lyapunov-like if and only if it is of the form L = L a + D, where a V and D is a (inner) derivation. In this situation, L a is the symmetric part of L and D is the skew-symmetric part of L. This result has already been observed in Prop. VIII.2.6, [7], perhaps, in a slightly different form. Here, we sketch its proof based on Lie algebraic ideas. A direct and elementary proof is presented in the Appendix.

8 Gowda, Tao, Ravindran: Complementarity properties 7 If D is a derivation, we have e td Aut(V ) Aut(K) for all t R and so D is Lyapunov-like. It turns out that every derivation D is also skew-symmetric (that is, D T + D = 0) or equivalently, D(e) = 0. Conversely, suppose L is Lyapunov-like and skew-symmetric. From L + L T = 0, we see that L and L T commute and so I = e t(l+lt) = e tl e tlt = e tl (e tl ) T t R. This says that e tl is an orthogonal transformation on V for all t R. Writing Orth(V ) for the group of all orthogonal transformations on V, we have e tl Orth(V ) Aut(K) t R. Now suppose that V is simple. Then, Orth(V ) Aut(K) = Aut(V ), see Pages 56 and 57 in [7]. Thus, e tl Aut(V ) t R. This means that L Der(V ), see Page 36 in [7]. Now when V is not simple, we can write V as a product of simple algebras V k, k =, 2,..., N. Then L can be written as a product of Lyapunov-like transformations L k on V k. It is easy to verify that each L k is skew-symmetric on V k. It follows from our previous discussion that each L k Der(V k ). Since the Jordan product in V is defined in a componentwise manner, we see that L Der(V ). So, every Lyapunov-like and skew-symmetric transformation is a derivation. Now suppose that L is any Lyapunov-like transformation and let a = L(e). Then L L a is Lyapunov-like and moreover, (L L a )(e) = 0. This implies that L L a is Lyapunov-like and skew-symmetric, and hence a derivation, say, D. This gives L = L a + D. We also note that the symmetric part of L, namely, (L+LT ) 2 is L a and the skew-symmetric part, namely, (L L T ) 2 is D. Remarks. It follows from the above theorem that if L is symmetric and Lyapunov-like, it must be of the form L a for some a V. 4. Lyapunov-like transformations on matrix algebras Let F denote the set of all real numbers/complex numbers/quaternions/octonions. Given any element p F, we write Re(p) for its real part and p for its conjugate. We note that quaternions are noncommutative but associative, while octonions are noncommutative and nonassociative. Still, for any three elements a, b and c in F, we have [4], Re(a) = Re(a), Re(ab) = Re(ba), and Re [a(bc)] = Re [(ab)c]. (4.2) For any A F n n, let tr(a) denote the sum of the diagonal elements of A. Then, for any three matrices A, B and C in F n n, see Prop. V.2., [7], Re tr(a) = Re tr(a ), Re tr(ab) = Re tr(ba), and Re tr (A(BC)) = Re tr ((AB)C), (4.3) where A is the conjugate transpose of A. Now, given A F n n, we define the Lyapunov transformation L A on Herm(F n n ) by The following extends a result of Damm [3]. L A (X) = AX + XA. Theorem 4.3 Let F denote real numbers, complex numbers, or quaternions. A linear transformation L on the Euclidean Jordan algebra Herm(F n n ) is Lyapunov-like if and only if there exists an A F n n such that L = L A.

9 8 Gowda, Tao, Ravindran: Complementarity properties Proof. Suppose that A F n n and consider X, Y Herm(F n n ) such that X 0, Y 0, and X, Y = 0. Then XY = Y X = 0. (This is well known for F = R or C; see Remark 3 in [7] for F = H.) Now, relying on the associativity in F, and using (4.3), L A (X), Y = Re tr(l A (X)Y ) = Re tr(axy + XA Y ) = Re tr(xa Y ) = Re tr(a Y X) = 0. This proves the Lyapunov-like property of L A on Herm(F n n ). Now for the converse. Suppose L is Lyapunov-like on Herm(F n n ). By the previous theorem, L = L A +D, where A Herm(F n n ) and D is a derivation. As D is inner, we can write m D = α i [L Ai, L Bi ], where α i are real numbers and A i, B i Herm(F n n ), i =, 2,..., m. Using the associativity of the ordinary matrix product of matrices in F n n, [L Ai, L Bi ] = L [Ai,B i]. It follows that D = L B, where B := m α i[a i, B i ] F n n. Hence, L = L A + L B = L A+B = L C, where C := A + B F n n. This completes the proof. We next show that a result of the previous type is not valid for matrices over octonions. Theorem 4.4 There exists A O 3 3 such that L A is not Lyapunov-like on Herm(O 3 3 ). Proof. By Remark 3 in [7], there exists a Jordan frame {E, E 2, E 3 } in Herm(O 3 3 ) such that E E 2 0. Now, E E 2 = 0 E E 2 + E 2 E = 0. Let p a b E E 2 := α q c. β γ r Then p ᾱ β E 2 E = (E E 2 ) = ā q γ. b c r From E E 2 + E 2 E = 0, we get Re(p) = Re(q) = Re(r) = 0, a + ᾱ = 0, b + β = 0, and c + γ = 0. We will construct an octonion matrix a a 2 a 3 A := a 2 a 22 a 23 a 3 a 32 a 33 such that L A (E ), E 2 0. As E E 2 0, some row of E E 2 is nonzero. Without loss of generality, assume that the first row [p a b] is nonzero; In this case, we take a ij = 0 for i {2, 3} and j {, 2, 3}. Then, using (4.3), and so Re tr((ae )E 2 ) = Re tr((a(e E 2 )) = Re tr((ae ) E 2 ) L A (E ), E 2 = Re tr(l A (E )E 2 ) = Re tr((ae )E 2 + (AE ) E 2 ) = 2 Re(a p + a 2 α + a 3 β). As [ p a b ] is nonzero, the vector [ p α β ] is also nonzero. Now, if p 0, we can take a = p, a 2 = a 3 = 0. Then L A (E ), E 2 = 2. A similar construction can be made if α or β is nonzero. Thus, A can be constructed so that L A (E ), E 2 0. This means that L A is not Lyapunov-like.

10 Gowda, Tao, Ravindran: Complementarity properties Lyapunov-like transformations on L n In this section, we give two characterizations of Lyapunov-like transformations on L n. Our second result, Theorem 4.6, can also be deduced from Theorem 4.2. Recalling that the underlying space of L n is R n (n > ), we fix a matrix A R n n and let J R n n be defined by J := diag(,,..., ). We recall the following from [5]: Lemma 4. The matrix A has the Z-property on L n if and only if there exists γ R such that γj (JA + A T J) is positive semidefinite on L n. As a consequence, we prove Theorem 4.5 The matrix A is Lyapunov-like on L n if and only if there exists β R such that βj + (JA + A T J) = 0. Proof. Suppose A is Lyapunov-like, in which case, A and A have the Z-property. Then by Lemma 4., there exist α and β such that αj (JA+A T J) and βj +(JA+A T J) are positive semidefinite. Therefore, αj (JA+A T J)+βJ +(JA+A T J) = (α+β)j is positive semidefinite. Thus, we have α = β. Now, βj (JA + A T J) = (βj + (JA + A T J)) is positive semidefinite and symmetric, hence βj + (JA + A T J) = 0. The if part is obvious because of the previous lemma. Here is another characterization of Lyapunov-like transformations on L n. Theorem 4.6 A matrix A on L n is a Lyapunov-like if and only if it is of the form [ a b T A = b D where a R, D R (n ) (n ), with D + D T = 2aI. ], (4.4) Proof. To see the if part, take x and y in L n such that 0 x y 0. Assuming that x and y are nonzero, we may write x = [ u ] T, y = [ u] T, where u = (see e.g., [2]). Then Ax, y = a u T Du = 0, where the last equality comes from D + D T = 2aI. This proves the Lyapunov-like property of A. Now for the only if part. Suppose the matrix A is Lyapunov-like and given by [ ] a b T A =. c D Putting x = [ u ] T, y = [ u ] T with u =, we see that 0 x y 0. Since A is Lyapunov-like, we have Ax, y = 0 and so Replacing u by u in (4.5), we have a + (b c) T u u T Du = 0. (4.5) a (b c) T u u T Du = 0. (4.6) The above two equations lead to (b c) T u = 0 for all u with u =. Thus, b = c. Now from the previous result, we have, βj + (JA+A T J) = 0 (for some β). This leads to β = 2a and D + D T = βi. Therefore, D + D T = 2aI. This completes the proof.

11 0 Gowda, Tao, Ravindran: Complementarity properties 5. The P-property of Lyapunov-like transformations Throughout this section, (in view of Theorem 3.2), we assume that the inner product is given by the trace inner product. This simplifies our proofs somewhat as c = for any primitive idempotent with respect to the trace inner product. Theorem 5. Let L be Lyapunov-like on V and L S. Then L P. Before giving a proof of this result, we prove several lemmas. In these lemmas, we fix a Jordan frame {e, e 2,...,e r } in V and consider the corresponding Peirce decomposition V = i j V ij given in Theorem 2.2. Throughout, we assume that L is Lyapunov-like. Lemma 5. For i j, let x ij V ij with x ij =. Then (a) x := e i + e j + 2x ij 0, (b) y := e i + e j 2x ij 0, and (c) x, y = 0. Proof. (a) In V, consider any d = d i e i + d ij 0. We note that 2d i d j d ij 2 for all i j, see, Exercise 7, Page 80, in [7]. Then, x, d = d i + d j + 2 x ij, d ij d i + d j 2 x ij d ij = d i + d j 2 d ij d i + d j 2 d i dj = ( d i d j ) 2 0. As K is self-dual, x 0. The proof of (b) is similar, and (c) is obvious. Lemma 5.2 The following hold: (a) i j, x ij V ij, x ij = L(x ij ), x ij = 2 ( L(e i), e i + L(e j ), e j ). (b) L(e i ), e i 0 i =, 2,...,r Tr(L) 0, where Tr(L) denotes the trace of the linear transformation L on V. Proof. (a) For a given x ij, let x and y be as in the previous lemma. Since L is Lyapunov-like, we have L(x), y = 0 = L(y), x. Now, L(x), y = 0 L(e i ), e i + L(e j ), e j 2 L(x ij ), x ij = 2 L(x ij ), e i + e j + 2 L(e i ) + L(e j ), x ij. (5.) L(y), x = 0 L(e i ), e i + L(e j ), e j 2 L(x ij ), x ij Adding these equations, we get the desired result. = 2 L(x ij ), e i + e j 2 L(e i ) + L(e j ), x ij. (5.2) (b) Now suppose L(e i ), e i 0 for all i =, 2,..., r. Recall that the trace of the linear transformation L on the (real) Hilbert space V is the sum of numbers of the form L(u), u, as u varies over any orthonormal basis in V. (This can be seen by writing the matrix representation of L with respect to an orthonormal basis and adding the diagonal elements of that matrix.) As V is an orthogonal direct sum of spaces V ij for i j, the union of orthonormal bases in various V ij s is an orthonormal basis in V. By our assumption and Item (a), the sum of numbers of the form L(u), u as u varies over an orthonormal basis in any V ij is nonpositive. Thus, the sum of these, namely, Tr(L), is also nonpositive. This completes the proof.

12 Gowda, Tao, Ravindran: Complementarity properties Lemma 5.3 Suppose i, k, and l are distinct and x kl V kl. Then L(e i ), x kl = 0 and L(x kl ), e i = 0. Proof. Without loss of generality, we assume that x kl =. Then e k + e l 2x kl 0 by Lemma 5., and e i, e k + e l 2x kl = 0 for i {k, l}. Since L is a Lyapunov-like transformation, we have L(e i ), e k + e l 2x kl = 0 L(e i ), x kl = 0. Similarly, L(e k + e l 2x kl ), e i = 0 L(x kl ), e i = 0. Lemma 5.4 Suppose i, j, k, and l are all distinct, x ij V ij, and x kl V kl. Then L(x ij ), x kl = 0. Proof. Without loss of generality, we assume that x ij = x kl =. Now, put u = e i +e j 2ε x ij and v = e k + e l 2δ x kl, where ε, δ {, }. Then 0 u v 0. Since L is Lyapunov-like, we have L(u), v = 0 and so 2δ L(e i + e j ), x kl + 2εδ L(x ij ), x kl 2ε L(x ij ), e k + e l = 0. (5.3) In (5.3), we replace ε by ε and δ by δ to get 2δ L(ei + e j ), x kl + 2εδ L(x ij ), x kl + 2ε L(x ij ), e k + e l = 0. (5.4) Now adding these two equations, we get L(x ij ), x kl = 0. Lemma 5.5 Corresponding to a Jordan frame {e, e 2,...,e r }, suppose there exists x = k x ie i such that x i 0 for all i =, 2,...,k and y = L(x) = k y ie i. Then L(e i ), z ij = 0, i {, 2,..., k}, j {k +, k + 2,...,r}, z ij V ij. Proof. Since the spaces V ij are mutually orthogonal and z ij V ij, we have L(x), z ij = y, z ij = 0. On expanding, we get x L(e ), z ij + x 2 L(e 2 ), z ij + + x i L(e i ), z ij + x i+ L(e i+ ), z ij + + x k L(e k ), z ij = 0. In the above equality, fix i {, 2,..., k}. By Lemma 5.3, L(e l ), z ij = 0 for all l i. For l = i, x i 0 and so L(e i ), z ij = 0. In our next lemma, we assume that V is simple and of rank 3. As V carries the trace inner product, we have, see Prop. IV..4 and Lemma IV.2.2 in [7], (x ij ) 2 = x ij 2 (e i +e j ), x ij (x ij y jl ) = 2 8 x ij 2 y jl and x ij y jl 2 = 8 x ij 2 y jl 2 (5.5) for all x ij V ij, y jl V jl, with i, j, and l distinct. Lemma 5.6 Suppose that the conditions of the previous lemma are in place. Assume that V is simple and rank (V ) 3. Then L(x ij ), y il = 0, for all i j {, 2,..., k}, l {k +, k + 2,...,r}, x ij V ij, and y il V il. Proof. Fix i j {, 2,..., k} and l {k +, k + 2,...,r}. Assume without loss of generality, x ij = = y il. Let x = e i + e j + 2ε x ij and y = αe i + αe j + e l 2ε α x ij + δ y il 2 2ε δ x ij y il, where ε, δ {, } and α R. Then, using the identities in (5.5), relations of the form x ij e i = 2 x ij and Theorem 2.2, we get 0 y 2 = y y = (2α )e i + (2α )e j + 2e l 2ε(2α )x ij + δ(2α + )y il 2 2εδ(2α + )x ij y il.

13 2 Gowda, Tao, Ravindran: Complementarity properties It is easy to verify that x, y 2 = 0. Since L is Lyapunov-like, we have L(x), y 2 = 0. This gives, after expanding and putting a = L(e i ), e i and b = L(e j ), e j, 0 = a(2α ) + b(2α2 + 2 ) 2ε(2α ) L(e i) + L(e j ), x ij + δ(2α + ) L(e i ) + L(e j ), y il 2 2εδ(2α + ) L(e i ) + L(e j ), x ij y il + (2α ) 2ε L(x ij ), e i + e j + 2 2ε L(x ij ), e l (4α 2 + ) L(x ij ), x ij (5.6) + 2εδ(2α + ) L(x ij ), y il 4δ(2α + ) L(x ij ), x ij y il. Replacing ε by ε and δ by δ in (5.6), we have 0 = a(2α ) + b(2α2 + 2 ) + 2ε(2α ) L(e i) + L(e j ), x ij δ(2α + ) L(e i ) + L(e j ), y il 2 2εδ(2α + ) L(e i ) + L(e j ), x ij y il (2α ) 2ε L(x ij ), e i + e j 2 2ε L(x ij ), e l (4α 2 + ) L(x ij ), x ij (5.7) + 2εδ(2α + ) L(x ij ), y il + 4δ(2α + ) L(x ij ), x ij y il. Since V ij V il V jl, L(e i ), x ij y il = 0 by Lemma 5.3 and L(e j ), x ij y il = 0 by Lemma 5.5. Adding equations (5.6) and (5.7), we get a(2α ) + b(2α2 + 2 ) (4α2 + ) L(x ij ), x ij + 2εδ(2α + ) L(x ij ), y il = 0. (5.8) Now replacing α by α and δ by δ in (5.8), we have a(2α ) + b(2α2 + 2 ) (4α2 + ) L(x ij ), x ij 2εδ( 2α + ) L(x ij ), y il = 0. (5.9) Subtracting equation (5.9) from (5.8), we get L(x ij ), y il = 0. Our previous lemmas lead to the following Proposition 5. Suppose V is simple with rank r and L is Lyapunov-like on V. Further suppose that corresponding to a Jordan frame {e, e 2,...,e r }, there exists x = k x ie i such that x i 0 for all i =, 2,...,k, and y = L(x) = k y i e i. Let W := V (e + e e k, ). Then L(W) W. Proof The result is obvious if k = or k = r. We assume that < k < r. This implies that r 3 and the previous Lemma is applicable. Consider the Peirce decomposition of any x W with respect to the Jordan frame {e, e 2,..., e k }: x = k x ie i + i<j k x ij. We write the Peirce decomposition of z := L(x) with respect to the Jordan frame {e, e 2,..., e r } in V : k x i L(e i )+ k L(x ij ) = ( z i e i + z ij )+( z il )+( z i e i + z il ). i<j k i<j k i k<k+ l r k+ k+ i<l r Now, taking the inner product of the term on the left-hand side of the above expression with z il for i k < k + l r or with z i e i for i k +, or with z il for k + i < l r, and using the previous lemmas, we deduce that the last two (grouped) terms on the right-hand side of the above expression are both zero. Thus, k L(x) = z i e i + z ij W. This completes the proof. i<j k Proof of Theorem 5.. We are given that L is Lyapunov-like on V and L S. To start with suppose that V is simple. Assume, if possible, there is a nonzero x such that

14 Gowda, Tao, Ravindran: Complementarity properties 3 x and y := L(x) operator commute, and x y 0. Then for some Jordan frame {e, e 2,..., e r } in V, x = x i e i and L(x) = y i e i, where x i y i 0 for all i =, 2,..., r. As x 0, we may assume that x i 0 for i k and x i = 0 for i > k, where k r. Then k x i L(e i ) = L(x) = y i e i, which leads to k x i L(e i ), e j = y j j =, 2,...,r, (5.0) as e j = for all j. Since L is Lyapunov-like, these yield y j = 0 for j k + so that Moreover, L(x) = L( k x i e i ) = k y i e i. L(e i ), e i = y i x i 0 i =, 2,...,k. (5.) Let W := V (e + e e k, ). Then by Proposition 5., L(W) W. Denote the restriction of L to W by L. Then L is Lyapunov-like on the (subalgebra) W. Also, the matrix representation of L on V = W W is of the form [ ] A B, 0 C where A is the matrix representation of L on W. Since L is Lyapunov-like and hence a Z- transformation, the S-property is equivalent to the positive stable property, see Theorems 6 and 7 in [5]. This implies, from the above matrix representation, that L also has the positive stable property and so its trace must be positive. However, (5.) implies that L (e i ), e i 0 for all i =, 2,...,k and by Lemma 5.2, applied to L and W, Tr(L ) 0. This is a contradiction. Hence no x exists, completing the proof of the theorem when V is simple. Now for the general case. Suppose that V is a product of simple algebras V k, k =, 2,...,N. By Theorem 4., we may write L as a product of linear transformations L k, where L k is Lyapunov-like on V k. Also, as L S, there is a 0 < d = (d, d 2,..., d n ) such that 0 < L(d) = (L (d ),...,L N (d N )). This means that L k S on V k. Now, by the first part of the proof, each L k has the P-property on V k. Since the P-property depends only on the Jordan product, it can be described in a componentwise manner. This means that the product of L k s, that is, L, has the P-property on V. This completes the proof of the theorem. Remarks. For any a V, L a is Lyapunov-like. In this case, the symmetric transformation L a has the P-property if and only if L a is strictly monotone. This can happen if and only if a > 0. A derivation D, while Lyapunov-like, can never be invertible because D(e) = 0. Hence it can never have any of the complementarity properties S, Q, or P. 6. Appendix In this Appendix, we present a proof of Theorem 4.2 without relying on Lie algebra techniques. We begin with a proposition. Proposition 6. The following are equivalent:

15 4 Gowda, Tao, Ravindran: Complementarity properties (i) L is a derivation. (ii) L is Lyapunov-like and L(e) = 0. (iii) L is Lyapunov-like and skew-symmetric. Proof. (i) (ii) : Let L be a derivation, {e, e 2,...,e r } be a Jordan frame in V, and i j. Using e i e j = 0 and writing 0 = L(e i e j ) = L(e i ) e j +e i L(e j ), we get, upon taking the inner product with e j, 0 = L(e i ) e j + e i L(e j ), e j = L(e i ), e j e j + L(e j ), e i e j = L(e i ), e j, where we used the properties e j e j = e j and x y, z = x, y z in V. This proves that for all i j, L(e i ), e j = 0, that is, L is Lyapunov-like. For any i, e i e i = e i. Applying L both sides, we get 2 L(e i ) e i = L(e i ) and 2 L(e i ) e i, e i = L(e i ), e i. This implies that L(e i ), e i = 0, i =, 2,..., r. So we have proved that when L is a derivation, for any Jordan frame {e, e 2,..., e r }, L(e i ), e j = 0 i, j =, 2,...,r. Fixing j and summing over i, we get L(e), e j = 0 for all j. As the Jordan frame is arbitrary, writing the spectral decomposition of any x as x = r x je j, we get L(e), x = 0. As x is arbitrary, this gives L(e) = 0. Thus we have proved that L is Lyapunov-like and L(e) = 0. (ii) (iii) : Let {e, e 2,...,e r } be any Jordan frame. Then L(e i ), e j = 0, i j. The condition L(e) = 0 implies that r L(e i) = 0 and hence L(e i ), e j = 0 even when i = j. Now for any x V, we have the spectral expansion x = r x ie i for some Jordan frame {e, e 2,..., e r } and eigenvalues x, x 2,..., x r. Then L(x), x = x i x j L(e i ), e j = 0. i,j This implies that L + L T = 0, i.e., L is skew-symmetric. (iii) (ii) : Assume that L is Lyapunov-like and skew-symmetric. Then for any Jordan frame {e, e 2,..., e r }, we have L(e i ), e j = 0 for all i and j. As in the last part of the proof of (i) (ii), we get L(e) = 0. (iii) (i) : Let D = L be Lyapunov-like and skew-symmetric. We first observe that D is a derivation provided D(x 2 ) = 2x D(x) x V. (6.) For, if this condition holds, then replacing x by x + λy and comparing coefficients of λ, we get D(x y) = x D(y) + y D(x). Now to prove (6.), take any x V and consider its spectral decomposition: x = x i e i. Then x 2 = x 2 i e i, and hence D(x 2 ) = x 2 i D(e i). Now, suppose that for all i and k l, Then 2e i D(e i ) = D(e i ) and e k D(e l ) + e l D(e k ) = 0. (6.2) 2x D(x) = 2( x i e i ) ( x j D(e j )) = 2( i = 2( i x 2 ie i D(e i )) + 2 i<j x 2 ie i D(e i )) x i x j (e i D(e j ) + e j D(e i )) = i x 2 i D(e i) = D(x 2 ).

16 Gowda, Tao, Ravindran: Complementarity properties 5 We now prove (6.2). Fix an index k, k r. We write the Peirce decomposition of D(e k ) as D(e k ) = x i e i + i<j x ij. Since D is skew symmetric, D(e i ), e i = 0 for i =,...,r. Since D is a Lyapunov-like, D(e k ), e j = 0 for k j. Thus, x i = 0 for i =,...,r. Hence D(e k ) = i<j x ij. Now, e k, e i + e j + 2x ij = 0 for k i, j and e i + e j + 2x ij 0 for i j (see Lemma 5.). Then D(e k ), e i + e j + 2x ij = 0. This implies D(e k ), x ij = 0. Thus, k D(e k ) = x ik + i= j=k+ Multiplying both sides of this equality by e k and using x ik e k = 2 x ik, etc., we get e k D(e k ) = 2 D(e k). This proves the first part of (6.2). Now for the second part. Based on the discussion above, we write the Peirce decomposition of D(e k ) for k =, 2,..., r: x kj. D(e ) = x 2 + x x r D(e 2 ) = y 2 + y y 2r D(e 3 ) = z 3 + z z 3r. =. k D(e k ) = p ik + i=... =. l D(e l ) = q il +. = i= Now, taking sum of these equations, we have. j=k+ j=l+ p kj q lj D(e r ) = w r + w 2r + + w r r. D(e) = (x 2 + y 2 ) + (x 3 + z 3 ) + + (p lk + q lk ) +. Since the grouped terms in D(e) belong to orthogonal Peirce spaces and D(e) = 0, we have p lk +q lk = 0. Since e k D(e l ) = 2 q lk and e l D(e k ) = 2 p lk, we have e k D(e l ) + e l D(e k ) = 2 (p lk + q lk ) = 0. This proves the second part of (6.2) and completes the proof of the proposition. Now to prove Theorem 4.2, let L be Lyapunov-like. Then with a = L(e), D := L L a is Lyapunov-like and D(e) = 0. Hence by the above Proposition, D is a derivation. It follows that L = L a + D completing the proof. Concluding Remarks. In this paper, we discussed the P-property of Z and Lyapunov-like transformations. We proved that for Lyapunov-like transformations, the P-property is equivalent to the S-property. In the case of Z-transformation, we do not have this full equivalence. So the problem of proving the equality Z S = Z P continues to be open. References [] Baker, A Matrix Groups, Springer-Verlag, London. [2] Cottle, R.W., J.-S. Pang, R.E. Stone The Linear Complementarity Problem. Academic Press, Boston. [3] Damm, T Positive groups on H n are completely positive. Linear Algebra Appl

17 6 Gowda, Tao, Ravindran: Complementarity properties [4] Dray, T., C.A. Manogue The octonionic eigenvalue problem. Adv. Appl. Clifford Alg [5] Elsner, L Quasimonotonie und ungleichungen in halbgeordneten Räumen. Linear Alg. Appl [6] Facchinei, F., J.-S. Pang Finite Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, New York. [7] Faraut, J., A. Korányi Analysis on Symmetric Cones. Oxford University Press, Oxford. [8] Ferris, M.C., O. Mangasarian, J.-S. Pang Complementarity: Applications, Algorithms and Extensions. Kluwer Academic Publishers, Boston. [9] Ferris, M.C., J.-S. Pang Complementarity and Variational Problems, State of the Art. Proceedings of the International Conference on Complementarity Problems, SIAM, Philadelphia. [0] Fiedler, M., V. Pták On matrices with non-positive off-diagonal elements and positive principle minors. Czechoslovak Math. J [] Gowda, M.S., T. Parthasarathy Complementarity forms of theorems of Lyapunov and Stein, and related results, Linear Algebra Appl [2] Gowda, M.S., Y. Song On semidefinite linear complementarity problems. Math. Prog. Series A [3] Gowda, M.S., R. Sznajder Some global uniqueness and solvability results for linear complementarity problems overs symmetric cones. SIAM J. Optimization [4] Gowda, M.S., R. Sznajder, J. Tao Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl [5] Gowda, M.S., J. Tao Z-transformations on proper and symmetric cones. Math. Prog. Series B [6] Koecher, M The Minnesota Notes on Jordan Algebras and Their Applications. Springer Lecture Notes in Mathematics, Berlin. [7] Moldovan, M.M., M.S. Gowda, Strict diagonal dominance and a Ger sgorin type theorem in Euclidean Jordan algebras. Linear Algebra Appl [8] Murty, K.G On the number of solutions to the complementarity problem and spanning properties of complementarity cones. Linear Algebra Appl [9] Schmieta, S.H., F. Alizadeh Extension of primal-dual interior point algorithms to symmetric cones. Math. Prog. Series A [20] Robinson, S.M Normal maps induced by linear transformations. Mathematics of Operations Research [2] Tao, J Some P-Properties for Linear Transformations on the Lorentz Cone. Ph.D. Thesis, University of Maryland Baltimore County, [22] Schneider, H., M. Vidyasagar, 970. Cross-positive matrices. SIAM Jour. Numer. Anal

Strict diagonal dominance and a Geršgorin type theorem in Euclidean

Strict diagonal dominance and a Geršgorin type theorem in Euclidean Strict diagonal dominance and a Geršgorin type theorem in Euclidean Jordan algebras Melania Moldovan Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland

More information

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

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 Gowda a, J. Tao

More information

Key words. Complementarity set, Lyapunov rank, Bishop-Phelps cone, Irreducible cone

Key words. Complementarity set, Lyapunov rank, Bishop-Phelps cone, Irreducible cone ON THE IRREDUCIBILITY LYAPUNOV RANK AND AUTOMORPHISMS OF SPECIAL BISHOP-PHELPS CONES M. SEETHARAMA GOWDA AND D. TROTT Abstract. Motivated by optimization considerations we consider cones in R n to be called

More information

Some characterizations of cone preserving Z-transformations

Some characterizations of cone preserving Z-transformations Annals of Operations Research manuscript No. (will be inserted by the editor) Some characterizations of cone preserving Z-transformations M. Seetharama Gowda Yoon J. Song K.C. Sivakumar This paper is dedicated,

More information

On the game-theoretic value of a linear transformation relative to a self-dual cone

On the game-theoretic value of a linear transformation relative to a self-dual cone On the game-theoretic value of a linear transformation relative to a self-dual cone M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland

More information

Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras

Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras positivity manuscript No. (will be inserted by the editor) Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras M. Seetharama Gowda Jiyuan Tao February

More information

ON POSITIVE SEMIDEFINITE PRESERVING STEIN TRANSFORMATION

ON POSITIVE SEMIDEFINITE PRESERVING STEIN TRANSFORMATION J. Appl. Math. & Informatics Vol. 33(2015), No. 1-2, pp. 229-234 http://dx.doi.org/10.14317/jami.2015.229 ON POSITIVE SEMIDEFINITE PRESERVING STEIN TRANSFORMATION YOON J. SONG Abstract. In the setting

More information

Z-transformations on proper and symmetric cones

Z-transformations on proper and symmetric cones Math. Program., Ser. B DOI 0.007/s007-007-059-8 FULL LENGTH PAPER Z-transformations on proper and symmetric cones Z-transformations M. Seetharama Gowda Jiyuan Tao Received: 6 February 006 / Accepted: 5

More information

On common linear/quadratic Lyapunov functions for switched linear systems

On common linear/quadratic Lyapunov functions for switched linear systems On common linear/quadratic Lyapunov functions for switched linear systems M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland, USA

More information

M. Seetharama Gowda a, J. Tao b & Roman Sznajder c a Department of Mathematics and Statistics, University of Maryland

M. Seetharama Gowda a, J. Tao b & Roman Sznajder c a Department of Mathematics and Statistics, University of Maryland This article was downloaded by: [University of Maryland Baltimore County] On: 03 April 2013, At: 09:11 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954

More information

Permutation invariant proper polyhedral cones and their Lyapunov rank

Permutation invariant proper polyhedral cones and their Lyapunov rank Permutation invariant proper polyhedral cones and their Lyapunov rank Juyoung Jeong Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA juyoung1@umbc.edu

More information

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland

More information

On the game-theoretic value of a linear transformation on a symmetric cone

On the game-theoretic value of a linear transformation on a symmetric cone On the game-theoretic value of a linear transformation ona symmetric cone p. 1/25 On the game-theoretic value of a linear transformation on a symmetric cone M. Seetharama Gowda Department of Mathematics

More information

A PREDICTOR-CORRECTOR PATH-FOLLOWING ALGORITHM FOR SYMMETRIC OPTIMIZATION BASED ON DARVAY'S TECHNIQUE

A PREDICTOR-CORRECTOR PATH-FOLLOWING ALGORITHM FOR SYMMETRIC OPTIMIZATION BASED ON DARVAY'S TECHNIQUE Yugoslav Journal of Operations Research 24 (2014) Number 1, 35-51 DOI: 10.2298/YJOR120904016K A PREDICTOR-CORRECTOR PATH-FOLLOWING ALGORITHM FOR SYMMETRIC OPTIMIZATION BASED ON DARVAY'S TECHNIQUE BEHROUZ

More information

Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras

Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras Lingchen Kong, Levent Tunçel, and Naihua Xiu 3 (November 6, 7; Revised December, 7) Abstract Recently, Gowda et al. [] established

More information

The eigenvalue map and spectral sets

The eigenvalue map and spectral sets The eigenvalue map and spectral sets p. 1/30 The eigenvalue map and spectral sets M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland

More information

Spectral sets and functions on Euclidean Jordan algebras

Spectral sets and functions on Euclidean Jordan algebras Spectral sets and functions on Euclidean Jordan algebras Juyoung Jeong Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA juyoung1@umbc.edu

More information

We describe the generalization of Hazan s algorithm for symmetric programming

We describe the generalization of Hazan s algorithm for symmetric programming ON HAZAN S ALGORITHM FOR SYMMETRIC PROGRAMMING PROBLEMS L. FAYBUSOVICH Abstract. problems We describe the generalization of Hazan s algorithm for symmetric programming Key words. Symmetric programming,

More information

L. Vandenberghe EE236C (Spring 2016) 18. Symmetric cones. definition. spectral decomposition. quadratic representation. log-det barrier 18-1

L. Vandenberghe EE236C (Spring 2016) 18. Symmetric cones. definition. spectral decomposition. quadratic representation. log-det barrier 18-1 L. Vandenberghe EE236C (Spring 2016) 18. Symmetric cones definition spectral decomposition quadratic representation log-det barrier 18-1 Introduction This lecture: theoretical properties of the following

More information

Polynomial complementarity problems

Polynomial complementarity problems Polynomial complementarity problems M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA gowda@umbc.edu December 2, 2016

More information

A continuation method for nonlinear complementarity problems over symmetric cone

A continuation method for nonlinear complementarity problems over symmetric cone A continuation method for nonlinear complementarity problems over symmetric cone CHEK BENG CHUA AND PENG YI Abstract. In this paper, we introduce a new P -type condition for nonlinear functions defined

More information

Error bounds for symmetric cone complementarity problems

Error bounds for symmetric cone complementarity problems to appear in Numerical Algebra, Control and Optimization, 014 Error bounds for symmetric cone complementarity problems Xin-He Miao 1 Department of Mathematics School of Science Tianjin University Tianjin

More information

Absolute value equations

Absolute value equations Linear Algebra and its Applications 419 (2006) 359 367 www.elsevier.com/locate/laa Absolute value equations O.L. Mangasarian, R.R. Meyer Computer Sciences Department, University of Wisconsin, 1210 West

More information

Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore,

More information

Mathematical Methods wk 2: Linear Operators

Mathematical Methods wk 2: Linear Operators John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

2.1. Jordan algebras. In this subsection, we introduce Jordan algebras as well as some of their basic properties.

2.1. Jordan algebras. In this subsection, we introduce Jordan algebras as well as some of their basic properties. FULL NESTEROV-TODD STEP INTERIOR-POINT METHODS FOR SYMMETRIC OPTIMIZATION G. GU, M. ZANGIABADI, AND C. ROOS Abstract. Some Jordan algebras were proved more than a decade ago to be an indispensable tool

More information

SCHUR IDEALS AND HOMOMORPHISMS OF THE SEMIDEFINITE CONE

SCHUR IDEALS AND HOMOMORPHISMS OF THE SEMIDEFINITE CONE SCHUR IDEALS AND HOMOMORPHISMS OF THE SEMIDEFINITE CONE BABHRU JOSHI AND M. SEETHARAMA GOWDA Abstract. We consider the semidefinite cone K n consisting of all n n real symmetric positive semidefinite matrices.

More information

Lecture 7: Positive Semidefinite Matrices

Lecture 7: Positive Semidefinite Matrices Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.

More information

Boolean Inner-Product Spaces and Boolean Matrices

Boolean Inner-Product Spaces and Boolean Matrices Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver

More information

POSITIVE MAP AS DIFFERENCE OF TWO COMPLETELY POSITIVE OR SUPER-POSITIVE MAPS

POSITIVE MAP AS DIFFERENCE OF TWO COMPLETELY POSITIVE OR SUPER-POSITIVE MAPS Adv. Oper. Theory 3 (2018), no. 1, 53 60 http://doi.org/10.22034/aot.1702-1129 ISSN: 2538-225X (electronic) http://aot-math.org POSITIVE MAP AS DIFFERENCE OF TWO COMPLETELY POSITIVE OR SUPER-POSITIVE MAPS

More information

The Q Method for Symmetric Cone Programming

The Q Method for Symmetric Cone Programming The Q Method for Symmetric Cone Programming Farid Alizadeh Yu Xia October 5, 010 Communicated by F. Potra Abstract The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton,

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

More information

Lecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min.

Lecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min. MA 796S: Convex Optimization and Interior Point Methods October 8, 2007 Lecture 1 Lecturer: Kartik Sivaramakrishnan Scribe: Kartik Sivaramakrishnan 1 Conic programming Consider the conic program min s.t.

More information

Linear Algebra: Matrix Eigenvalue Problems

Linear Algebra: Matrix Eigenvalue Problems CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

More information

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra 1.1. Introduction SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that

More information

Geometric Mapping Properties of Semipositive Matrices

Geometric Mapping Properties of Semipositive Matrices Geometric Mapping Properties of Semipositive Matrices M. J. Tsatsomeros Mathematics Department Washington State University Pullman, WA 99164 (tsat@wsu.edu) July 14, 2015 Abstract Semipositive matrices

More information

STRUCTURE AND DECOMPOSITIONS OF THE LINEAR SPAN OF GENERALIZED STOCHASTIC MATRICES

STRUCTURE AND DECOMPOSITIONS OF THE LINEAR SPAN OF GENERALIZED STOCHASTIC MATRICES Communications on Stochastic Analysis Vol 9, No 2 (2015) 239-250 Serials Publications wwwserialspublicationscom STRUCTURE AND DECOMPOSITIONS OF THE LINEAR SPAN OF GENERALIZED STOCHASTIC MATRICES ANDREAS

More information

Chapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of

Chapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of Chapter 2 Linear Algebra In this chapter, we study the formal structure that provides the background for quantum mechanics. The basic ideas of the mathematical machinery, linear algebra, are rather simple

More information

Clifford Algebras and Spin Groups

Clifford Algebras and Spin Groups Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of

More information

Fundamentals of Engineering Analysis (650163)

Fundamentals of Engineering Analysis (650163) Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is

More information

Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control

Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the

More information

Department of Social Systems and Management. Discussion Paper Series

Department of Social Systems and Management. Discussion Paper Series Department of Social Systems and Management Discussion Paper Series No. 1262 Complementarity Problems over Symmetric Cones: A Survey of Recent Developments in Several Aspects by Akiko YOSHISE July 2010

More information

Foundations of Matrix Analysis

Foundations of Matrix Analysis 1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the

More information

Matrices A brief introduction

Matrices A brief introduction Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Matrices Semester 1, 2014-15 1 / 44 Definitions Definition A matrix is a set of N real or complex

More information

Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore,

More information

Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX

Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX September 2007 MSc Sep Intro QT 1 Who are these course for? The September

More information

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to 1.1. Introduction Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that

More information

1. General Vector Spaces

1. General Vector Spaces 1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

Linear Algebra. Workbook

Linear Algebra. Workbook Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

More information

Elementary linear algebra

Elementary linear algebra Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

More information

The Jordan Algebraic Structure of the Circular Cone

The Jordan Algebraic Structure of the Circular Cone The Jordan Algebraic Structure of the Circular Cone Baha Alzalg Department of Mathematics, The University of Jordan, Amman 1194, Jordan Abstract In this paper, we study and analyze the algebraic structure

More information

Summer School: Semidefinite Optimization

Summer School: Semidefinite Optimization Summer School: Semidefinite Optimization Christine Bachoc Université Bordeaux I, IMB Research Training Group Experimental and Constructive Algebra Haus Karrenberg, Sept. 3 - Sept. 7, 2012 Duality Theory

More information

Algebra II. Paulius Drungilas and Jonas Jankauskas

Algebra II. Paulius Drungilas and Jonas Jankauskas Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive

More information

Analysis of Control Systems on Symmetric Cones

Analysis of Control Systems on Symmetric Cones Analysis of Control Systems on Symmetric Cones Ivan Papusha Richard M. Murray Abstract It is well known that exploiting special structure is a powerful way to extend the reach of current optimization tools

More information

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2 Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

More information

Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,

Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, 2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for

More information

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88 Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

More information

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

Maxwell s equations. Daniel Henry Gottlieb. August 1, Abstract

Maxwell s equations. Daniel Henry Gottlieb. August 1, Abstract Maxwell s equations Daniel Henry Gottlieb August 1, 2004 Abstract We express Maxwell s equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current,

More information

Some Properties of the Augmented Lagrangian in Cone Constrained Optimization

Some Properties of the Augmented Lagrangian in Cone Constrained Optimization MATHEMATICS OF OPERATIONS RESEARCH Vol. 29, No. 3, August 2004, pp. 479 491 issn 0364-765X eissn 1526-5471 04 2903 0479 informs doi 10.1287/moor.1040.0103 2004 INFORMS Some Properties of the Augmented

More information

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability... Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................

More information

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2 EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

More information

Properties of Solution Set of Tensor Complementarity Problem

Properties of Solution Set of Tensor Complementarity Problem Properties of Solution Set of Tensor Complementarity Problem arxiv:1508.00069v3 [math.oc] 14 Jan 2017 Yisheng Song Gaohang Yu Abstract The tensor complementarity problem is a specially structured nonlinear

More information

A Primal-Dual Second-Order Cone Approximations Algorithm For Symmetric Cone Programming

A Primal-Dual Second-Order Cone Approximations Algorithm For Symmetric Cone Programming A Primal-Dual Second-Order Cone Approximations Algorithm For Symmetric Cone Programming Chek Beng Chua Abstract This paper presents the new concept of second-order cone approximations for convex conic

More information

ON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION

ON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION Annales Univ. Sci. Budapest., Sect. Comp. 33 (2010) 273-284 ON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION L. László (Budapest, Hungary) Dedicated to Professor Ferenc Schipp on his 70th

More information

Largest dual ellipsoids inscribed in dual cones

Largest dual ellipsoids inscribed in dual cones Largest dual ellipsoids inscribed in dual cones M. J. Todd June 23, 2005 Abstract Suppose x and s lie in the interiors of a cone K and its dual K respectively. We seek dual ellipsoidal norms such that

More information

Jordan-algebraic aspects of optimization:randomization

Jordan-algebraic aspects of optimization:randomization Jordan-algebraic aspects of optimization:randomization L. Faybusovich, Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN, 46545 June 29 2007 Abstract We describe a version

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

Linear algebra 2. Yoav Zemel. March 1, 2012

Linear algebra 2. Yoav Zemel. March 1, 2012 Linear algebra 2 Yoav Zemel March 1, 2012 These notes were written by Yoav Zemel. The lecturer, Shmuel Berger, should not be held responsible for any mistake. Any comments are welcome at zamsh7@gmail.com.

More information

An LMI description for the cone of Lorentz-positive maps II

An LMI description for the cone of Lorentz-positive maps II An LMI description for the cone of Lorentz-positive maps II Roland Hildebrand October 6, 2008 Abstract Let L n be the n-dimensional second order cone. A linear map from R m to R n is called positive if

More information

ENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS. 1. Introduction

ENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS. 1. Introduction Trends in Mathematics Information Center for Mathematical Sciences Volume 6, Number 2, December, 2003, Pages 83 91 ENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS SEUNG-HYEOK KYE Abstract.

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

SOME STABILITY RESULTS FOR THE SEMI-AFFINE VARIATIONAL INEQUALITY PROBLEM. 1. Introduction

SOME STABILITY RESULTS FOR THE SEMI-AFFINE VARIATIONAL INEQUALITY PROBLEM. 1. Introduction ACTA MATHEMATICA VIETNAMICA 271 Volume 29, Number 3, 2004, pp. 271-280 SOME STABILITY RESULTS FOR THE SEMI-AFFINE VARIATIONAL INEQUALITY PROBLEM NGUYEN NANG TAM Abstract. This paper establishes two theorems

More information

5.6. PSEUDOINVERSES 101. A H w.

5.6. PSEUDOINVERSES 101. A H w. 5.6. PSEUDOINVERSES 0 Corollary 5.6.4. If A is a matrix such that A H A is invertible, then the least-squares solution to Av = w is v = A H A ) A H w. The matrix A H A ) A H is the left inverse of A and

More information

Linear Algebra 2 Spectral Notes

Linear Algebra 2 Spectral Notes Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex

More information

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra. DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1

More information

Lecture Note 5: Semidefinite Programming for Stability Analysis

Lecture Note 5: Semidefinite Programming for Stability Analysis ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State

More information

Chapter 3 Transformations

Chapter 3 Transformations Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases

More information

Linear Algebra Massoud Malek

Linear Algebra Massoud Malek CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

More information

Review of Vectors and Matrices

Review of Vectors and Matrices A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P

More information

Some parametric inequalities on strongly regular graphs spectra

Some parametric inequalities on strongly regular graphs spectra INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 Some parametric inequalities on strongly regular graphs spectra Vasco Moço Mano and Luís de Almeida Vieira Email: vascomocomano@gmailcom Faculty of

More information

EIGENVALUES AND EIGENVECTORS 3

EIGENVALUES AND EIGENVECTORS 3 EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices

More information

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 12 NO. 1 PAGE 1 (2019) Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space Murat Bekar (Communicated by Levent Kula ) ABSTRACT In this paper, a one-to-one

More information

Linear Algebra 1. M.T.Nair Department of Mathematics, IIT Madras. and in that case x is called an eigenvector of T corresponding to the eigenvalue λ.

Linear Algebra 1. M.T.Nair Department of Mathematics, IIT Madras. and in that case x is called an eigenvector of T corresponding to the eigenvalue λ. Linear Algebra 1 M.T.Nair Department of Mathematics, IIT Madras 1 Eigenvalues and Eigenvectors 1.1 Definition and Examples Definition 1.1. Let V be a vector space (over a field F) and T : V V be a linear

More information

Criss-cross Method for Solving the Fuzzy Linear Complementarity Problem

Criss-cross Method for Solving the Fuzzy Linear Complementarity Problem Volume 118 No. 6 2018, 287-294 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Criss-cross Method for Solving the Fuzzy Linear Complementarity Problem

More information

Real Symmetric Matrices and Semidefinite Programming

Real Symmetric Matrices and Semidefinite Programming Real Symmetric Matrices and Semidefinite Programming Tatsiana Maskalevich Abstract Symmetric real matrices attain an important property stating that all their eigenvalues are real. This gives rise to many

More information

BOOK REVIEWS 169. minimize (c, x) subject to Ax > b, x > 0.

BOOK REVIEWS 169. minimize (c, x) subject to Ax > b, x > 0. BOOK REVIEWS 169 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 1, January 1993 1993 American Mathematical Society 0273-0979/93 $1.00+ $.25 per page The linear complementarity

More information

Kernel Method: Data Analysis with Positive Definite Kernels

Kernel Method: Data Analysis with Positive Definite Kernels Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University

More information

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the

More information

APPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.

APPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2. APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product

More information

Linear Algebra. Min Yan

Linear Algebra. Min Yan Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................

More information

Lecture 2: Linear operators

Lecture 2: Linear operators Lecture 2: Linear operators Rajat Mittal IIT Kanpur The mathematical formulation of Quantum computing requires vector spaces and linear operators So, we need to be comfortable with linear algebra to study

More information

Control System Analysis on Symmetric Cones

Control System Analysis on Symmetric Cones Submitted, 015 Conference on Decision and Control (CDC) http://www.cds.caltech.edu/~murray/papers/pm15-cdc.html Control System Analysis on Symmetric Cones Ivan Papusha Richard M. Murray Abstract Motivated

More information

Mathematics Department Stanford University Math 61CM/DM Inner products

Mathematics Department Stanford University Math 61CM/DM Inner products Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector

More information

12x + 18y = 30? ax + by = m

12x + 18y = 30? ax + by = m Math 2201, Further Linear Algebra: a practical summary. February, 2009 There are just a few themes that were covered in the course. I. Algebra of integers and polynomials. II. Structure theory of one endomorphism.

More information

2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian

2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian

More information

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,

More information