On the inverse of a special Schur complement

The inverse of a Schur complement is a very useful tool in many algorithms for the computation of the matrix inversion. In this paper we study the inverse of a special Schur complement CD−1B. We proved that always exist some X and Y such that XDY is a inverse of CD−1B. Furthermore, using minimal rank properties, we give some explicit expressions for X, Y and the inverse of CD−1B. Numerical examples also given.