Factorization of CP-rank-$3$ completely positive matrices

Czechoslovak Mathematical Journal

Michal Křížek¹  Jan Brandts² 

[1] Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic;Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

关键词: completely positive matrix;    cp-rank;    factorization;    discrete maximum principle;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A=BB^\top$. 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.

【 授权许可】

Unknown