Generalized principal pivot transform

The generalized principal pivot transform is a generalization of the principal pivot transform to the singular case, using the Moore–Penrose inverse. In this article we study some properties of the generalized principal pivot transform. We prove that the Moore–Penrose inverse of a range-symmetric, almost skew-symmetric matrix is almost skew-symmetric. It is shown that the generalized principal pivot transform preserves the rank of the symmetric part of a matrix under some conditions.