Determinantal criteria for total positivity

A matrix is totally positive (respectively, strictly totally positive) if all its minors are nonnegative (respectively, positive). In this paper, we revisit the main determinantal criteria for these matrices and provide two new ones: one of them for the total positivity of m×n matrices with any n consecutive rows linearly independent and another one to check if an n×n totally positive matrix has all minors of order less than k (k n) positive.