Quasi-LDU factorization of nonsingular totally nonpositive matrices

Let A=(aij)∈Rn×n be a nonsingular totally nonpositive matrix. In this paper we describe some properties of these matrices when a11 = 0 and obtain a characterization in terms of the quasi-LDU factorization of A, where L is a block lower triangular matrix, D is a diagonal matrix and U is a unit upper triangular matrix.