Relaxing the Nonsingularity Assumption for Intervals of Totally Nonnegative Matrices

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard partial order are considered. It is proven that if the two bound matrices of such a matrix interval are totally nonnegative and satisfy certain conditions, then all matrices from this interval are totally nonnegative and satisfy these conditions, too, hereby relaxing the nonsingularity condition in the former paper [M. Adm and J. Garloff. Intervals of totally nonnegative matrices. Linear Algebra Appl., 439:3796--3806, 2013.].

Total Nonnegativity and Stable Polynomials

We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point V of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if V is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix A preserves stability of polynomials if and only if A is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya–Schur theory of Borcea and Brändén.