Matrix monotonicity and concavity of the principal pivot transform

TL;DR

This paper establishes that the generalized principal pivot transform is matrix monotone and convex under minimal conditions, extending previous results by relaxing hypotheses and using a variational principle.

Contribution

It proves the matrix monotonicity and convexity of the generalized principal pivot transform under the weakest known hypotheses, improving upon recent prior work.

Findings

The generalized principal pivot transform is matrix monotone under minimal hypotheses.

The principal pivot transform is matrix convex on positive semi-definite matrices with the same kernel.

The proof relies on a new variational principle for the transform.

Abstract

We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the L\"owner ordering, under minimal hypotheses. This improves on the recent results of J. E. Pascoe and R. Tully-Doyle, Monotonicity of the principal pivot transform, Linear Algebra Appl. 662 (2022) 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 on matrices for which monotonicity holds 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 convex 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.