On Schur complements of sign regular matrices of order k

The issue regarding Schur complements of sign regular matrices is rather subtle. It is known that the class of totally nonnegative matrices is not closed under arbitrary Schur complementation. In this paper, we demonstrate how Schur complements of sign regular matrices of order k are sign regular of a certain order. In particular, some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries.