Factorization of CP-rank-3 completely positive matrices

TL;DR

This paper presents a finite, exact algorithm for factorizing symmetric positive semi-definite matrices with cp-rank 3, providing a method to determine such matrices and exploring related polynomial orthogonality conditions.

Contribution

The paper introduces a novel finite algorithm to factorize cp-rank-3 matrices, advancing the understanding of their structure and properties.

Findings

Algorithm successfully identifies cp-rank-3 matrices

Failure indicates the matrix does not have cp-rank 3

Application to polynomial orthogonality problems

Abstract

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A=BB^T. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.