Preservers of Completely Positive Matrix Rank for Inclines

Abstract

A real symmetric matrix A is called completely positive if there exists a nonnegative real \(n\times k\) matrix B such that \(A = BB^{t}\). The smallest value of k for all possible choices of nonnegative matrices B is called the CP-rank of A. We extend the ideas of complete positivity and the CP-rank to matrices whose entries are elements of an incline in a similar way. We classify maps on the set of \(n \times n\) symmetric matrices over certain inclines which strongly preserve CP-rank-1 matrices as well as maps which preserve CP-rank-1 and CP-rank-k. The result suggests that there is a certain standard class of solutions for CP-rank preserver problems on incline matrices.