Sign patterns of inverse doubly nonnegative matrices and inverse completely positive matrices

Abstract

We identify all possible {+,−,0} sign patterns of inverse doubly nonnegative (DNN) matrices, and of all inverse completely positive (CP) matrices. We prove that all inverses of DNN realizations of a connected graph share the same {+,−,0} sign pattern if and only if the graph is bipartite, and the same holds in CP case. In the DNN case, the characterization generalizes a result of Roy and Xue (Linear Algebra and its Applications 610 (2021) 480–487) [14] regarding the {+,−} sign pattern of inverse DNN matrices, where + denotes a nonnegative entry, and the second result answers a question left open there. We also consider the reverse question: which {+,−,0} sign patterns of inverse DNN/CP matrices determine uniquely the graph of their originating DNN/CP matrix. We answer the question in the DNN case, but the CP case is still open.