Conditions for the positivity of determinants

Abstract

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 sufficiet conditions for an arbitrary matrix A whose (undirected) graph is subordinate to a tree, to be a P-matrix. If, in additionA 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.

^∗Work supported in part by Office of Naval Research contract N00014-90-J-1739 and NSF grant DMS 92-00899.

^†Work supported by NSF Grant DMS-8901860 and NSF Grant DMS-9306357

^‡Work exported by NSFRC Grant 6-53121.

^∗Work supported in part by Office of Naval Research contract N00014-90-J-1739 and NSF grant DMS 92-00899.

^†Work supported by NSF Grant DMS-8901860 and NSF Grant DMS-9306357

^‡Work exported by NSFRC Grant 6-53121.

Notes

^∗Work supported in part by Office of Naval Research contract N00014-90-J-1739 and NSF grant DMS 92-00899.

^†Work supported by NSF Grant DMS-8901860 and NSF Grant DMS-9306357

^‡Work exported by NSFRC Grant 6-53121.