The maximal cp-rank of rank k completely positive matrices

Matrices with High Completely Positive Semidefinite Rank

A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by positive semidefinite matrices (of any size d). The smallest such d is called the completely positive semidefinite rank of M , and it is an open question whether there exists an upper bound on this number as a function of the matrix size. We show that if such an upper bound exists, it has to be a...

Completely Positive Semidefinite Rank

An n×n matrix X is called completely positive semidefinite (cpsd) if there exist d×d Hermitian positive semidefinite matrices {Pi}i=1 (for some d ≥ 1) such that Xij = Tr(PiPj), for all i, j ∈ {1, . . . , n}. The cpsd-rank of a cpsd matrix is the smallest d ≥ 1 for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the c...

On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

We show that the maximal cp-rank of n×n completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of n×n completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of 5×5 matrices equals six, which proves the famous Drew-JohnsonLoewy conjecture (1994) for matrices of this order. In addition we present a s...

On higher rank numerical hulls of normal matrices

‎In this paper‎, ‎some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated‎. ‎A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given‎. ‎Moreover‎, ‎using the extreme points of the numerical range‎, ‎the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$‎, ‎where $A_1...

Commuting maps on rank-k matrices