On the necessity and sufficiency of \(PLUS\) factorizations (Q1779261)

A \(PLUS\) factorization for an arbitrary nonsingular \(n\times n\) matrix \(A\) has the form \(A=PLUS\), where \(P\) is a permutation matrix, \(L\) is a unit lower triangular matrix, \(U\) is an upper triangular matrix whose diagonal entries are prescribed as long as the determinant is equal to that of \(A\) up to a possible sign adjustment, and \(S\) is a unit lower triangular matrix of which all but \(n-1\) off-diagonal entries are zeros and the positions of those \(n-1\) entries are also flexibly customizable. The authors show that the necessary condition for the existence of a \(PLUS\) factorization of a matrix \(A\) as given by \textit{P. Hao} [ibid. 382, 135--154 (2004; Zbl 1050.15012)] is not sufficient and they find a sufficient condition for such a factorization.