🤖 AI Summary
This work addresses the long-standing issue of inconsistency in generalized matrix inverses under nonsingular diagonal transformations—a problem particularly critical in robotics, tracking, and control systems where physical unit sensitivity is paramount. The paper introduces a novel generalized inverse that, for the first time, achieves invariance under arbitrary nonsingular diagonal transformations, thereby rigorously preserving the physical units of state-space variables. This new inverse, together with the Moore–Penrose and Drazin inverses, forms a complete triad encompassing the principal linear system transformations. The framework is further extended to unit-consistent matrix factorizations. Grounded in matrix analysis theory, the authors develop a new algebraic construction method, successfully applied across multiple engineering domains, providing a robust mathematical foundation for unit-preserving modeling and significantly advancing the theoretical completeness of generalized inverses.
📝 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.