A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations
This addresses a fundamental issue in robotics, tracking, and control systems by enabling unit preservation under state space transformations, though it appears incremental as it builds on existing generalized inverses.
The paper introduces a new generalized matrix inverse that remains consistent under arbitrary nonsingular diagonal transformations, solving a longstanding open problem in fields like robotics and control systems. It complements existing inverses like Drazin and Moore-Penrose to complete a trilogy covering standard linear system transformations.
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.