The paper deals with the stability of a convex hull of matrices and a positive cone of matrices.
The author obtains a sufficient condition for a matrix family such that the stability of a finitely many well-chosen matrices guarantees stability of the convex hull and positive cone of the whole matrix family. In some special cases the condition is also necessary. On the other hand, he establishes a relationship between the real parts of the eigenvalues and the matrix measure, showing that if the matrix measure is properly chosen, then it is possible to obtain stability criteria which are equivalent to testing the real parts of the eigenvalues for negativity.
Finally, by using the matrix measure, the author finds a result for the stability of time-invariant linear systems with parameter variations representing the unmodeled dynamics.