A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations

TL;DR

This paper introduces a new generalized matrix inverse that maintains consistency under diagonal transformations, addressing a longstanding problem with applications in robotics, tracking, and control systems.

Contribution

It presents a novel inverse that is consistent with diagonal transformations, complementing existing inverses and completing the family of analytically-important linear system transformations.

Findings

The new inverse preserves units under state space transformations.

It complements the Drazin and Moore-Penrose inverses.

Examples demonstrate its application in matrix decompositions.

Abstract

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore-Penrose inverse (which is consistent with respect to unitary/orthonormal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically-important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions and examples of their use are described.