A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations
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.
Cite
@article{arxiv.2604.00049,
title = {A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations},
author = {Jeffrey Uhlmann},
journal= {arXiv preprint arXiv:2604.00049},
year = {2026}
}
Comments
This reflects the 2018 SIMAX publication. (The 1604.08476 preprint has a comment saying that its content is contained in the SIMAX paper, but the two are quite distinct.)