Ordering of subclasses of trees by algebraic connectivity is a very active area of research. Let G = (V,E) be a simple undirected graph on n vertices. The Laplacian matrix of G is the n × n matrix L (G) = D (G) − A (G) where A (G) is the adjacency matrix and D (G) is the diagonal matrix of vertex degrees. It is well known that L (G) is a positive semidefinite matrix and that (0, e) is an eigenp...

Abstract. The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a 1 graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by 2 G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive 3 semidefinite zero forcing number Z+(G) is introduced, and ...

We consider a general bitangential interpolation problem for matrix Schur functions and focus mainly on the case when the associated Pick matrix is singular (and positive semidefinite). Descriptions of the set of all solutions are given in terms of special linear fractional transformations which are obtained using two quite different approaches. As applications of the obtained results we consid...

The low-rank solutions of continuous and discrete Lyapunov equations are of great importance but generally difficult to achieve in control system analysis and design. Fortunately, Mesbahi and Papavassilopoulos [On the rank minimization problems over a positive semidefinite linear matrix inequality, IEEE Trans. Auto. Control, Vol. 42, No. 2 (1997), 239-243] showed that with the semidefinite cone...

We show how the zero structure of a basis of the null space of a positive semidefinite matrix can be exploited to very accurately compute its Cholesky factorization. We discuss consequences of this result for the solution of (constrained) linear systems and eigenvalue problems. The results are of particular interest if A and the null space basis are sparse.

We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as max-cut. Other applications include max-min eigenvalue problems and relaxations for the stable set problem.

We consider partial differential operators H = − div(C∇) in divergence form on R with a positive-semidefinite, symmetric, matrix C of real L∞-coefficients and establish that H is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.

We present a general semidefinite relaxation scheme for general n-variate quartic polynomial optimization under homogeneous quadratic constraints. Unlike the existing sum-of-squares (SOS) approach which relaxes the quartic optimization problems to a sequence of (typically large) linear semidefinite programs (SDP), our relaxation scheme leads to a (possibly nonconvex) quadratic optimization prob...

The convex cone of n × n completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for n ≤ 4 only, every DNN matrix is CPP. In this paper, we investigate the difference between 5×5...

We examine the problem of approximating a positive, semidefinite matrix Σ by a dyad xxT , with a penalty on the cardinality of the vector x. This problem arises in the sparse principal component analysis problem, where a decomposition of Σ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, th...