Full rank factorization in quasi-LDU form of totally nonpositive rectangular matrices

Let A=(aij)∈Rn×m be a totally nonpositive matrix with rank(A)=r⩽min{n,m} and a11=0. In this paper we obtain a characterization in terms of the full rank factorization in quasi-LDU form, that is, A=L˜DU where L˜∈Rn×r is a block lower echelon matrix, U∈Rr×m is a unit upper echelon totally positive matrix and D∈Rr×r is a diagonal matrix, with rank(L˜)=rank(U)=rank(D)=r. We use this quasi-LDU decomposition to construct the quasi-bidiagonal factorization of A. Moreover, some properties about these matrices are studied.