2023/02/08 by Kenneth Beard, Beard, Kenneth, Aaron Welters +1 · 1 citation
Mathematics · #15A09 #15A10 #15A15 #15A39 #15B48 #15B57 #47A56 #47B44 #47L07 #90C33 #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2302.04293
openalex publication_date 2023/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.