Positive Maps and Separable Matrices

Positive Linear Maps and Spreads of Matrices

Positive linear maps and spreads of matrices-II

It is shown how positive linear maps can be used to obtain lower bounds for spreads of normal matrices. Some known results and some new ones are obtained in this way. AMS classification: 15A45, 15A60, 47A12,

Piecewise Separable Matrices

Piecewise separable matrices are introduced as a natural generalization of semiseparable and profile-low-rank matrices. It is shown that all matrices from this class possess linear matrix-by-vector complexity.

Almost Separable Matrices

An m × n matrix A with column supports {Si} is k-separable if the disjunctions ⋃ i∈K Si are all distinct over all sets K of cardinality k. While a simple counting bound shows that m > k log2 n/k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker definition of ‘almost k-separability’, whic...

POSITIVE MAPS, ABSOLUTELY MONOTONIC FUNCTIONS AND THE REGULARIZATION OF POSITIVE DEFINITE MATRICES By

We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J, 26], Christensen and Ressel [Trans. Amer. Math. Soc., 243], and others, where similar problems were studied when the function is applied to all elements, in...