A real matrix is called totally nonnegative and totally positive if all of its minors are
nonnegative and positive, respectively. Such matrices arise in a remarkable variety of ways in
mathematics and many areas of its applications. It has a long time been conjectured by Juergen
Garloff, that each matrix in a matrix interval with respect to the checkerboard partial order is
nonsingular totally nonnegative if the two corner matrices are so. The Garloff’s Conjecture was
affirmatively answered for the totally positive matrices and some subclasses of the totally nonnegative
In this talk, we present a condensed form of the Cauchon Algorithm which provides an efficient criterion
for total nonnegativity of a given matrix and gives an optimal determinantal test for total nonnegativity.
We briefly report on the way in which Garloff’s Conjecture was settled. These and related results evoke
the (open) question whether the interval property holds for general nonsingular sign regular matrices.
I will conclude my talk by reporting on the usefulness of such matrices in studying real stable polynomials,
i.e., polynomials having their zeros in the open left half of the complex plane, and presenting a sufficient
condition for an interval family of polynomials to be stable by using intervals of totally nonnagtive matrices.