A matrix A is called completely positive if it can be decomposed as A = BBT with an entrywise nonnegative matrix B. The set of all such matrices is a convex cone which plays a role in certain optimization problems. A characterization of the interior of this cone is provided.

An n × n complex sign pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the complex sign pattern class of S such that its characteristic polynomial is f(λ). If S is a spectrally arbitrary complex sign pattern matrix, and no proper subpattern of S is spectrally arbitrary, then S is a minimal sp...

Let A be a finite-dimensional power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f on S is called a subnorm if f(a) > 0 for all 0 = a ∈ S, and f(αa) = |α|f(a) for all a ∈ S and α ∈ F. If in addition, S is closed under raising to powers, then a subnorm f is said to be stable if there exists a positive c...

For every completely positive matrix A, cp-rankA ≥ rankA. Let cp-rankG be the maximal cp-rank of a CP matrix realization of G. Then for every graph G on n vertices, cp-rankG ≥ n. In this paper the graphs G on n vertices for which equality holds in the last inequality, and graphs G such that cp-rankA = rankA for every CP matrix realization A of G, are characterized.

In this paper, the perturbation problems of A (2) T,S are considered. By virtue of the gap between subspaces, we derive conditions that make the perturbation of A (2) T,S stable when T, S and A have suitable perturbations. At the same time, explicit formulas for perturbation of A (2) T,S and new results on perturbation bounds are obtained.

The variety C(3, n) of commuting triples of n × n matrices over C is shown to be irreducible for n = 7. It had been proved that C(3, n) is reducible for n ≥ 30, but irreducible for n ≤ 6. Guralnick and Omladič have conjectured that it is reducible for n > 7.

We solve the matrix equation XA+AX = 0, where A ∈ C is an arbitrary given square matrix, and we compute the dimension of its solution space. This dimension coincides with the codimension of the tangent space of the congruence orbit of A. Hence, we also obtain the (real) dimension of congruence orbits in C. As an application, we determine the generic canonical structure for congruence in C and a...

In this paper, necessary and sufficient conditions are stated for the Craig-Sakamoto equation det(I − sA − tB) = det(I − sA)det(I − tB) to hold for all scalars s, t ∈ C. Moreover, spectral properties for matrices A and B that satisfy this equation are investigated.