A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity

TL;DR

This paper introduces a new generalized matrix inverse that maintains rank and similarity transformation consistency, addressing limitations of existing inverses like the Drazin inverse.

Contribution

A novel generalized inverse is proposed that preserves rank and is consistent with similarity transformations, improving upon existing methods.

Findings

The new inverse preserves the rank of the original matrix.

It provides consistency with respect to similarity transformations.

Experiments indicate a need for more numerically stable algorithms.

Abstract

There has recently been renewed recognition of the need to understand the consistency properties that must be preserved when a generalized matrix inverse is required. The most widely known generalized inverse, the Moore-Penrose pseudoinverse, provides consistency with respect to orthonormal transformations (e.g., rotations of a coordinate frame), and a recently derived inverse provides consistency with respect to diagonal transformations (e.g., a change of units on state variables). Another well-known and theoretically important generalized inverse is the Drazin inverse, which preserves consistency with respect to similarity transformations. In this paper we note a limitation of the Drazin inverse is that it does not generally preserve the rank of the linear system of interest. We then introduce an alternative generalized inverse that both preserves rank and provides consistency with respect to similarity transformations. Lastly we provide an example and discuss experiments which suggest the need for algorithms with improved numerical stability.