A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations
arXiv:2604.00049v1 Announce Type: cross 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
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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.
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.)
Subjects:
Numerical Analysis (math.NA); Robotics (cs.RO)
Cite as: arXiv:2604.00049 [math.NA]
(or arXiv:2604.00049v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2604.00049
arXiv-issued DOI via DataCite
Journal reference: SIAM Journal on Matrix Analysis, Vol. 239, No. 2, pp. 781-800, 2018
Submission history
From: Jeffrey Uhlmann [view email] [v1] Mon, 30 Mar 2026 15:06:02 UTC (22 KB)
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