Consecutive-column and -row properties of matrices and the loewner-neville factorization

We say that a square matrix over a ring with identity has the consecutive-column property if for all k, all its relevant submatrices having k consecutive rows and the first k columns are invertible. Similarly, the consecutive-row (CR) is defined. We show that, analogously to the commutative case for totally positive matrices, a matrix has both CC and CR properties if and only if it admits a certain Loewner-Neville-type factorization (with invertible entries); this factorization is unique. Since the result is proved for matrices in such generality, it holds also for block matrices over a field with all blocks square. Explicit both-ways formulae are found between two sets of parameters: the Loewner-Neville coefficients in the factorization and the Schur complements of relevant submatrices in relevant submatrices larger by one. We also show that for lower-triangular matrices, the CC property is preserved by inversion.