Matrix monotonicity and concavity of the principal pivot transform

Abstract

We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the Löwner ordering, under minimal hypotheses. This improves on the recent results of Pascoe and Tully-Doyle (2022) [69] in two ways. First, we use the “generalized” principal pivot transform, where matrix inverses in the classical definition of the principal pivot transform are replaced with Moore-Penrose pseudoinverses. Second, the hypotheses they used to prove the monotonicity is relaxed and, in particular, we find the weakest hypotheses possible for which it can be true. We also prove the principal pivot transform is a matrix concave function on positive semi-definite matrices that have the same kernel (and, in particular, on positive definite matrices). Our proof is a corollary of a minimization variational principle for the principal pivot transform.