Structured matrices and inverses

TL;DR

This paper explores the properties of structured matrices with small displacement rank, demonstrating that various types of inverses, including Moore-Penrose, retain structure, enabling efficient computation and storage.

Contribution

It extends the understanding of structured matrices by showing that their inverses, including Moore-Penrose inverses, are also structured under practical definitions.

Findings

Inverse of structured matrices is also structured.

Moore-Penrose inverse of rank-deficient matrices is structured.

Supports fast inversion and reduced storage for structured matrices.

Abstract

A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced storage requirements. According to two definitions of displacement structure of practical interest, it is shown here that several types of inverses are also structured, including the Moore-Penrose inverse of rank-deficient matrices.