Non-symmetric matrix and unified theory of least squares

Unified theory of least squares was given by Rao (197l, 1972, 1973) and Rao and Mitra (197l). In the generalized inverse (g-inverse) matrix (R + XGX')− involved in unified theory of least squares, G is any symmetric matrix. We give the properties with matrix quadratic form of non-symmetric matrix, and extend symmetric matrix G to an arbitrary square matrix. i.e. G is not necessarily symmetric only if matrix XGX' is symmetric.