Principal minors and diagonal similarity of matrices (Q1074686)

This is a continuation of the paper of \textit{D. J. Hartfiel} and the author [ibid. 58, 147-167 (1984; Zbl 0541.15005)]. The present paper deals with the relation between diagonal similarity of two \(n\times n\) matrices A and B, with entries in a field F, and the property of A and B having equal corresponding principal minors. The main result is the following: Suppose \(n\geq 4\), A is irreducible, and for every partition of \(\{\) 1,2,...,n\(\}\) into subsets \(\alpha\), \(\beta\) with \(| \alpha | \geq 2\), \(| \beta | \geq 2\) either rank A[\(\alpha\) \(| \beta]\geq 2\) or rank A[\(\beta\) \(| \alpha]\geq 2\). If A and B have equal corresponding principal minors, of all orders, then B or \(B^ T\) is diagonally similar to A.