On generalized inverses of a block in a partitioned matrix (Q1109098)

Let M and G be partitioned square matrices consisting of the blocks, A, x, u', c, and B, y, v', d, respectively; c and d denote scalars, x, y, u and v are vectors of appropriate size, and the prime denotes transpose. Provided that G is a generalized inverse of M the author gives a set of conditions for B to be a generalized inverse of A with some specific properties (e.g. \(B=BAB\), AB is symmetric, B is the Moore-Penrose inverse of A). Computational formulae for different generalized inverses of A are also derived. Particular attention is paid to the cases where M and G are symmetric \((x=u\), \(y=v)\) or nonnegative definite. The results can be applied to re-estimation problems of parameters \(\beta =(\beta_ 1,\beta_ 2,...,\beta_ m)'\) of a linear regression model \(Y=X\beta +\epsilon\), \(E(\epsilon)=0\), \(Var(\epsilon)=\sigma\) 2V when one of the parameters is to be dropped.