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A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations

Jeffrey Uhlmann

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.

A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations

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.

Paper Structure

This paper contains 9 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Left and Right UC Generalized Inverses
  3. UC Generalized Inverse for Elemental-Nonzero Matrices
  4. The Fully General Unit-Consistent Generalized Inverse
  5. Unit-Consistent/Invariant Matrix Decompositions
  6. Discussion
  7. Uniqueness of the UC Inverse
  8. Alternative Constructions
  9. Implementations

Key Result

Lemma 2.2

Existence of a left-diagonal scale function according to Definition defdl is established by instantiating $\hbox{${\cal D}_{\hbox{\tiny L}}$}[\hbox{${\bf A}$}]=\hbox{${\bf D}$}$ with where $\hbox{${\bf A}$}(i,:)$ is row $i$ of $\hbox{${\bf A}$}$ and $\hbox{$\lVert \cdot \rVert$}$ is a fixed unitary-invariant vector normThe unitary-invariant norm used here is necessary only because of the imposed

Theorems & Definitions (22)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • ...and 12 more