Conditions for the Positivity of Determinants

Consider a matrix with positive diagonal entries, which is similar via a positive diagonal matrix to a symmetric matrix, and whose signed directed graph has the property that if a cycle and its symmetrically placed complement have the same sign, then they are both positive. We provide sufficient conditions so that A be a P–matrix, that is, a matrix whose principal minors are all positive. We further provide sufficient conditions for an arbitrary matrix A whose (undirected) graph is subordinate to a tree, to be a P–matrix. If, in addition, A is sign symmetric and its undirected graph is a tree, we obtain necessary and sufficient conditions that it be a P–matrix. We go on to consider the positive semi–definiteness of symmetric matrices whose graphs are subordinate to a given tree and discuss the convexity of the set of all such matrices.