A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914435759603712 |
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| author | Uhlmann, Jeffrey |
| author_facet | Uhlmann, Jeffrey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00049 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations Uhlmann, Jeffrey Numerical Analysis Robotics 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. |
| title | A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations |
| topic | Numerical Analysis Robotics |
| url | https://arxiv.org/abs/2604.00049 |