A NOTE ON SOME CLASSES OF G-MATRICES

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). 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 this note, a characterization of G(D-1,D-2) is obtained and some properties of these G-classes are exhibited, such as conditions for equality of two G-classes. It is shown that G(D-1,D-2) has two or four connected components in M-n and that G(n) = boolean OR(D1,D2) G(D-1,D-2), the set of all n x n G-matrices, has two connected components in M-n. Sign patterns of the G-classes are also examined.