2018/07/09 by Jeffrey Uhlmann · 1 citation
Computer Science · Mathematics · #Matrix Theory and Algorithms #Neural Networks and Applications #Advanced Optimization Algorithms Research #Moore–Penrose pseudoinverse #Mathematics #Rank (graph theory) #Orthonormal basis #Inverse #Consistency (knowledge bases) #Drazin inverse #Generalized inverse #Similarity (geometry) #Matrix (chemical analysis) #Applied mathematics #Computer science #Combinatorics #Discrete mathematics #Artificial intelligence
paper · doi:10.1109/lcsys.2018.2854240
openalex created_date 2018/05/07 · openalex publication_date 2018/07/09 · openalex updated_date 2026/08/05
There has recently been renewed recognition of the need to understand the consistency properties that must be preserved when a generalized matrix inverse is required. The most widely known generalized inverse, the Moore-Penrose pseudoinverse, provides consistency with respect to orthonormal transformations (e.g., rotations of a coordinate frame), and a recently derived inverse provides consistency with respect to diagonal transformations (e.g., a change of units on state variables). Another well-known and theoretically important generalized inverse is the Drazin inverse, which preserves consistency with respect to similarity transformations. In this letter we note that the Drazin inverse suffers a significant practical limitation in that it does not generally preserve the rank of the linear system of interest. We then introduce an alternative generalized inverse that both preserves rank and provides consistency with respect to similarity transformations. We discuss practical implementation considerations and demonstrate with an example.