On nonsingularity of a polytope of matrices

The nonsingularity problem of a polytope of real matrices and its relation to the (robust) stability problem is considered. This problem is investigated by using the Bernstein expansion of the determinant function. Here we adapt the known Bernstein algorithm for checking the positivity of a multivariate polynomial on a box to the nonsingularity problem. It is shown that for a family of Z-matrices the positive stability problem is equivalent to the nonsingularity if this family has a stable member. It is established that the stability of the convex hull of real matrices A1, A2, … , Ak is equivalent to the nonsingularity of the convex hull of matrices A1, A2, … , Ak, jI if A1 is stable.