On matrices having equal corresponding principal minors (Q794741)

A pair of \(n\times n\) matrices A and B over a field F satisfies property \({\mathcal D}\) if B or \(B^ t\) is diagonally similar to A. If A and B satisfy property \({\mathcal D}\), then they have equal corresponding principal minors of all orders. In this paper the authors modify the converse problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property \({\mathcal D}\). These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility.