On nonsingular sign regular matrices

An m×n matrix A is sign regular if, for each k (1 k min{m,n}), all k×k submatrices of A have determinant with the same nonstrict sign. The zero pattern of nonsingular sign regular matrices is analyzed. It is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature. A matrix is totally nonpositive if all its minors are nonpositive. A test for recognizing nonsingular totally nonpositive matrices is also provided.