We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that stability implies that there exists a quadratic Lyapunov function on the state space, although this is in general infinite dimensional. We give an explicit parametrization of a finite-dimensional subset of the cone of Lyapunov functions using positive semid...

We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.

X. Zhan has conjectured that the spectral radius of the Hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. We prove Zhan’s conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, Kronecker products, and the spectral radius.

Spectrahedra are linear sections of the cone of positive semidefinite matrices which, as convex bodies, generalize the class of polyhedra. In this paper we investigate the problem of recognizing when a spectrahedron is polyhedral. We generalize and strengthen results of [M. V. Ramana, Polyhedra, spectrahedra, and semidefinite programming, in Topics in Semidefinite and Interior-Point Methods, Fi...

We consider the special class of semidefinite linear programs (IV P ) maximize traceCX subject to L A(X) U, where C,X,L, U are symmetric matrices, A is a (onto) linear operator, and denotes the Löwner (positive semidefinite) partial order. We present explicit representations for the general primal and dual optimal solutions. This extends the results for standard linear programming that appeared...

A semidefinite program (SDP) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints, and the condition that the unknown matrix is positive semidefinite. Standard techniques for solving SDP’s require O(n) operations per iteration. We introduce subspace algorithms that greatly reduce the cost os sol...

In this paper, we obtain some matrix inequalities in Löwner partial ordering for Khatri-Rao products of positive semidefinite Hermitian matrices. Furthermore, we generalize the Oppenheim’s inequality, with which we will improve some recent results.

Engineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real numbers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of po...

In their paper, Recht and Ré have presented conjectures and consequences of noncommutative variants of the arithmetic mean-geometric mean (AM-GM) inequality for positive definite matrices. Let A1, . . . , An be a collection of positive semidefinite matrices and i1, . . . , ik be random indices in {1, . . . , n}. To avoid symmetrization issues that arise since matrix products are non-commutative...

We give elementary proofs of the fact that the Loewner matrices [ f(pi)−f(pj) pi−pj ] corresponding to the function f(t) = t on (0,∞) are positive semidefinite, conditionally negative definite, and conditionally positive definite, for r in [0, 1], [1, 2], and [2, 3], respectively. We show that in contrast to the interval (0,∞) the Loewner matrices corresponding to an operator convex function on...