On a matrix identity associated with generalized least squares (Q908990)

For reduced generalized least squares equations with a heteroscedastic dispersion matrix V, the following two matrix identities are proved: \[ V^{-1}-V^{-1}X(X^ TV^{-1}X)^-X^ TV^{-1}=\{[I-X(X^ TX)^- X^ T]V[I-X(X^ TX)^--X^ T]\} \] and \[ V^{-1}(I-Q)=[(I-P)V(I- P)]^+ \] where \(A^-\) and \(A^+\) denote an arbitrary and the Moore- Penrose generalized inverse respectively and \(A^ T\) denotes the transpose of the matrix A. Here P is the projection \(P=X(X^ TX)^-X^ T\) where X is an (n\(\times p)\) matrix and V a positive definite dispersion matrix. Illustration of these identities are developed for the regression model \(Ey=X\beta +Z\gamma,\) \(Dy=V\) where E and D are the expectation and dispersion operators for the random (n\(\times 1)\) vector y.