$LDU$ Factorization of Nonsingular Totally Nonpositive Matrices
Rafael Cantó, Plamen Koev, Beatriz Ricarte, Ana M. Urbano · SIAM Journal on Matrix Analysis and Applications · 2008
An $n \times n$ real matrix A is said to be (totally negative) totally nonpositive if every minor is (negative) nonpositive. In this paper, we study the properties of a totally nonpositive matrix and characterize the case of a nonsingular totally nonpositive matrix A, with $a_{11}< 0$ in terms of its $LDU$ factorization ($L(U)$) is a unit lower- (upper-) triangular matrix, respectively, and D is a diagonal matrix). This characterization allows us to significantly reduce the number of minors to be checked in order to decide the total nonpositivity of a nonsingular matrix with a negative (1,1) entry.