Field of values of matrix polytopes

The tool of field of values (also known as the classical numerical range) is used to recover most results available in the literature and to obtain some new one s concerning Hurwitz and Schur stability of matrix polytopes. Some facts obtained by an application of the elementary properties of field of values are as follows. If the vertex matrices have polygonal field of values, then the matrix polytope is Hurwitz and Schur stable if and only if the vertex matrices are Hurwitz and Schur stable, respectively. If the polytope is nonnegative and the symmetric part of each vertex matrix is Schur stable, then the polytope is Schur stable. For polytopes with spectral vertex matrices, Schur stability of vertices is necessaryand sufficient for the Schur stability of the polytope.