Several conditions are derived for a polytope of Matrices to belong to P-matrices or M-matrices. These matrix classes play a significant role in many fields of systems science including econometrics, numerical analysis, nonlinear network theory, control theory, and so on. Since a polytope of matrices is a typical way to represent matrices whose entries are uncertain, the results obtained will render themselves useful tools, when one tries to introduce the robustness concept into the above theories.
The first major result of the present paper is a sufficient condition, which is expressed as a simple constraint to each vertex matrix. This is in force both for P-matrices and M-matrices. However, much simpler condition is available especially for M-matrices. As the second result, necessary and sufficient conditions are given for a convex combination of two matrices to be in the two matrix classes. Thus, we can obtain exact conditions when the number of vertex matrix is only two. All these results can be carried over to interval matrices, because an interval matrix can be expressed as a polytope of matrices whose entries are either of the end points of given intervals. Some comments are made on the parallelism of the properties between a polytope of matrices and that of polynomials.