This paper considers inclusion relations between the following four classes of real square matrices:
The diagonally stable matrices consisting of those matrices A such that is positive definite for some positive definite diagonal matrix D, the positive stable matrices consisting of those matrices A such that is positive definite for some positive definite matrix X, the P-matrices consisting of those A all of whose principal minors are positive, and the semi-positive matrices consisting of all A such that for some
In particular, it is shown that if A is a P-matrix and if the nondirected graph of A is a forest, then A is diagonally stable. (The nondirected graph of an matrix A is the graph whose vertices are 1,2,..., and whose edges are the pairs i,, , for which either or is nonzero.) This result, in view of previously known results, yields for those matrices whose nondirected graph is a forest the relations where means "is contained in" and the absence of an implication implies a counterexample. Partial results and some open questions are given for real spectra matrices and for a class of matrices called -matrices.