A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity

Abstract

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 paper we note a limitation of the Drazin inverse is 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. Lastly we provide an example and discuss experiments which suggest the need for algorithms with improved numerical stability.

Included simulation results and revised text in preparation for journal submission

Cited by in corpus (4)

• On the Relative Gain Array (RGA) with Singular and Rectangular Matrices
• A Generalized Matrix Inverse with Applications to Robotic Systems
• On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems
• Expression of a Real Matrix as a Difference of a Matrix and its Transpose Inverse