Completely positive matrices over Boolean algebras and their CP-rank

Abstract

Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [︀ n 2 /4 ]︀ . In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition, we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms. Definition 1.4. [11] A semiring is a set S together with two operations ⊕ and ⊗ and two distinguished elements0, 1 in S with 0 ≠ 1, such that 1. (S, ⊕, 0) is a commutative monoid, 2. (S, ⊗, 1) is a monoid, 3. ⊗ is both left and right distributive over ⊕, 4. The additive identity 0 satisfies the property r ⊗ 0 = 0 ⊗ r = 0, for all r ∈ S.

Bounds for the Completely Positive Rank of a Symmetric Matrix over a Tropical Semiring

“…Let M n (S) denote the semiring of all n×n matrices over the semiring S. Over the tropical semiring T, Cartwright and Chan [6] proved that max n, n 2 4 is the tight upper bound for the CP-rank of a completely positive matrix A ∈ M n (T). Over the Boolean semiring and the max-min semiring, the same inequality was proved by Mohindru [11] and Shitov [14].…”

Cholesky decomposition of matrices over commutative semirings

“…We use the following notion of positive semidefiniteness of matrices over semirings, which was defined in [9,8].…”