Balaji Ramamurthy

In this paper, we investigate the Lipschitz continuity of the solution map in semidefinite linear complementarity problems. For a monotone linear transformation defined on the space of real symmetric n × n matrices, we show that the Lipschitz continuity of the solution map implies the globally uniquely solvable (GUS)-property. For Lyapunov transformations with the Q-property, we prove that the Lipschitz continuity of the solution map is equivalent to the strong monotonicity property. For the double-sided multiplicative transformations, we show that the Lipschitz continuity of the solution map implies the GUS-propeity. © 2005 INFORMS.

Let G be a strongly connected and balanced directed graph. We define the resistance (Formula presented.) between any two vertices i and j of G by using the Moore–Penrose inverse of the Laplacian matrix of G and define the resistance matrix by (Formula presented.). This generalizes the resistance in the undirected case. In this paper, we show that R is a non-negative matrix and obtain an expression to compute the inverse, determinant and cofactor sums of R.

Let (Formula presented.) be the second-order cone in (Formula presented.), where n ≥ ~3. Given a vector (Formula presented.) and an n × n matrix G, the second order cone linear complementarity problem SOLCP(G, q) is to find a vector (Formula presented.) such that (Formula presented.) The matrix G is said to have the Q-property if SOLCP(G, q) has a solution for all (Formula presented.). An n × n matrix G is called a cone automorphism if (Formula presented.). In this paper, we obtain a simple characterization for the Q-property of a cone automorphism. This says that G has the Q-property if and only if zero is the only solution to SOLCP(G, 0).

Given a Euclidean Jordan algebra (V,o,〈.,.〉) with the (corresponding) symmetric cone K, a linear transformation L:V→V and qεV, the linear complementarity problem LCP(L,q) is to find a vector xεV such thatxεK, y:=L(x)+qεK and xy=0. To investigate the global uniqueness of solutions in the setting of Euclidean Jordan algebras, the P-property and its variants of a linear transformation were introduced in Gowda et al. (2004) [3] and it is shown that if LCP(L,q) has a unique solution for all qεV, then L has the P-property but the converse is not true in general. In the present paper, when (V,o,〈,〉) is the Jordan spin algebra, we show that LCP(L,q) has a unique solution for all qεV if and only if L has the P-property and L is positive semidefinite on the boundary of K.

Let G = (V, E) be a strongly connected and balanced digraph with vertex set V = {1, …, n}. The classical distance dij between any two vertices i and j in G is the minimum length of all the directed paths joining i and j. The resistance distance (or, simply the resistance) between any two vertices i and j in V is defined by rij:= lii† + l†jj − 2l†ij, where l†pq is the (p, q)th entry of the Moore-Penrose inverse of L which is the Laplacian matrix of G. In practice, the resistance rij is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between i and j is always less than or equal to the classical distance, i.e., rij ≤ dij . However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.

Let K⊆IRn denote the second-order cone. Given an n×n real matrix M and a vector q∈IRn, the second-order cone linear complementarity problem SOLCP(M,q) is to find a vector x∈IRn such thatx∈K,y:=Mx+q∈KandyTx=0. We say that M∈Q if SOLCP(M,q) has a solution for all q∈IRn. An n×n real matrix A is said to be a Z-matrix with respect to K iff:x∈K,y∈KandxTy=0  ⟹xTMy≤0. Let ΦM(q) denote the set of all solutions to SOLCP(M,q). The following results are shown in this paper: • If M∈Z∩Q, then ΦM is Lipschitz continuous if and only if M is positive definite on the boundary of K.• If M is symmetric, then ΦM is Lipschitz continuous if and only if M is positive definite.

Euclidean distance matrices ((Formula presented.)) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ((Formula presented.) s) that include (Formula presented.) s. Each (Formula presented.) is an entry-wise nonnegative matrix. A (Formula presented.) is not symmetric unless it is an (Formula presented.). By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore–Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices.

Let K ⊃ ^Rn be the n-dimensional Lorentz cone. Given an n×n matrix M and q ε ^Rn, the Lorentz-cone linear complementarity problem LCLCP(M,q) is to find an x ε^Rn that satisfiesx ε K,y:=Mx+q ε Kand^yTx=0. We show that if M is a Z-matrix with respect to K, then M is positive stable if and only if LCLCP(M,q) has a non-empty finite solution set for all q ε ^Rn.

Let (Formula presented.) denote the wheel graph having n-vertices. If i and j are any two vertices of (Formula presented.), define (Formula presented.) Let D be the (Formula presented.) matrix with (Formula presented.) entry equal to (Formula presented.). The matrix D is called the distance matrix of (Formula presented.). Suppose (Formula presented.) is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an (Formula presented.) matrix (Formula presented.) and a rank one matrix (Formula presented.) such that (Formula presented.) Here, (Formula presented.) is positive semidefinite, (Formula presented.) and all row sums are equal to zero.

The P-property of the following two Z-transformations with respect to the positive semidefinite cone is characterized: (i) I-S, where S: Sn×n→Sn×n is a nilpotent linear transformation, (ii) I-L-1 A, where LA is the Lyapunov transformation defined on Sn×n by LA(X) = AX + XAT. (Here Sn×n denotes the space of all symmetric n×n matrices and I is the identity transformation.).