We consider ergodic coherent MIMO channels. We characterize the optimal input distribution for various fading matrix distributions. First, we describe how symmetries in the fading matrix distribution are preserved as symmetries in the optimal input covariance and thus yield to specification of the optimal input. We will see that group structures and notion of commutant appear as key elements. S...

The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i, j)th entry (for i 6= j) is zero if i and j are not adjacent in G, is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extende...

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence y ∈ R n + whose moment matrix M(y) is positive semidefinite and has finite rank r is the sequence of moments of an r-atomic nonnegative measure μ on Rn. We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools use...

The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but non-CP...

We give an overview of cone optimization software, with special attention to the differences between the existing packages. We assume the reader is familiar with the theory and algorithms of cone optimization, thus technical details are kept at a minimum. We also outline current research trends and area of potential improvement. 1 Problem description Conic optimization solvers target problems o...

In this paper we describe General Covariance Union (GCU) and show that solutions to GCU and the Minimum Enclosing Ellipsoid (MEE) problems are equivalent. This is a surprising result because GCU is defined over positive semidefinite (PSD) matrices with statistical interpretations while MEE involves PSD matrices with geometric interpretations. Their equivalence establishes an intersection betwee...

In this paper, block distance matrices are introduced. Suppose F is a square block matrix in which each block is a symmetric matrix of some given order. If F is positive semidefinite, the block distance matrix D is defined as a matrix whose (i, j)-block is given by Dij = Fii+Fjj−2Fij . When each block in F is 1 × 1 (i.e., a real number), D is a usual Euclidean distance matrix. Many interesting ...

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do this we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator which ground state is determined by the lowest ...

Let Ln be the n-dimensional second order cone. A linear map from Rm to Rn is called positive if the image of Lm under this map is contained in Ln. For any pair (n,m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality (LMI) that describes this cone. Namely, we show that its dual cone, the cone of Lorentz-Lorentz separable elements, is a section o...