Total Positivity, $QR$ Factorization, and Neville Elimination

A well-known characterization of nonsingular totally positive matrices is improved: Only the sign of minors with consecutive initial rows or consecutive initial columns has to be checked. On the other hand, a new characterization of such matrices by their $QR$ factorization is obtained. As in other recent papers of the authors, Neville elimination plays an essential role. A well-known characterization of nonsingular totally positive matrices is improved: Only the sign of minors with consecutive initial rows or consecutive initial columns has to be checked. On the other hand, a new characterization of such matrices by their $QR$ factorization is obtained. As in other recent papers of the authors, Neville elimination plays an essential role.