The first author and Hall defined recently a G-matrix as a real non-singular matrix A such that there exist diagonal matrices D-1 and D-2 for which (A(T))(-1) = D(1)AD(2). The class of G-matrices was shown to possess interesting properties. In this paper, some new characterizations are found and extensions to rectangular and complex matrices are discussed. (C) 2012 Elsevier Inc. All rights reserved.