It is known that symmetric interval matrices are Hurwitz stable or positive definite if and only if all the vertex symmetric matrices are so. This paper generalizes this fact in two directions. Interval matrices dealt with here are such a class of matrices that have only real eigenvalues, thus including symmetric interval matrices. The other generalization is made by taking account of eigenvalue distribution in a more general manner. Namely, we consider the situation where, for some real k, a certain number of eigenvalues are located in the left of k while the remaining ones in the right of k. An imaginary axis couterpart of the obtained result is also provided.