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/orthogonal 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.