Monotonicity of the principal pivot transform

TL;DR

This paper proves that the principal pivot transform preserves the positivity of the imaginary part of matrices and establishes its monotonicity through Hermitian square representations.

Contribution

It demonstrates the matrix monotonicity of the principal pivot transform and provides new Hermitian square representations for its imaginary part and derivative.

Findings

Principal pivot transform maps matrices with positive imaginary part to similar matrices.

The transform is shown to be matrix monotone.

Hermitian square representations are established for the imaginary part and derivative.

Abstract

We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the principal pivot transform is matrix monotone by establishing Hermitian square representations for the imaginary part and the derivative.