Some results on the intersection of g-classes of matrices

The rich collection of G-matrices originated in a 2012 paper by Fiedler and Hall. Let M-n be the set of all n x n real matrices. A nonsingular matrix A is an element of M-n is called a G-matrix if there exist nonsingular diagonal matrices D-1 and D-2 such that A(-T) = D(1)AD(2), where A-T denotes the transpose of the inverse of A. For fixed nonsingular diagonal matrices D-1 and D-2, let G(D-1, D-2) = {A is an element of M-n : A(-T )= D(1)AD(2)}, which is called a G-class. In more recent papers, G-classes of matrices were studied. The purpose of this present work is to find conditions on D-1, D-2, D-3 and D-4 such that the G-classes G(D-1, D-2) and G(D-3, D-4) have finite nonempty intersection or empty intersection. A main focus of this work is the use of the diagonal matrix D = (D3D1-1/21)-D-1/2 . In the case that all the D-i are n x n diagonal matrices with positive diagonal entries, complete characterizations of the G-classes are obtained for the intersection questions.