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

@article{Uhlmann2019ARG, title={A Rank-Preserving Generalized Matrix Inverse for Consistency With Respect to Similarity}, author={J. Uhlmann}, journal={IEEE Control Systems Letters}, year={2019}, volume={3}, pages={91-95} }

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… Expand

#### Figures and Topics from this paper

#### 10 Citations

N A ] 2 S ep 2 02 0 Canonical Tensor Scaling

- 2020

A 1992 algorithm by Rothblum & Zenios [1], which we will refer to as the RZ algorithm, computes a positive diagonal scaling of an arbitrary matrix so that the products of the nonzero normed elements… Expand

Examining a Mixed Inverse Approach for Stable Control of a Rover

- 2020

In this paper we significantly extend previous work on methods for ensuring that control system behavior is invariant with respect to units chosen for critical state variables when intermediate… Expand

A Generalized Matrix Inverse with Applications to Robotic Systems

- Computer Science, Engineering
- ArXiv
- 2018

This paper shows that a recently introduced generalized matrix inverse permits performance consistency to be rigorously guaranteed in control systems that require solutions to underdetermined and/or overdetermined systems of equations. Expand

On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems

- Computer Science, Engineering
- ArXiv
- 2021

This paper considers the application of the MP-RGA to a realistic system to assess whether or not the choice of unit, which in this case relates to temperature, affects the choices of input-output pairings determined by the resulting RGA matrix, and shows that it does, confirming that unit-sensitivity of the MPs undermines its rigorous use for MIMO controller design. Expand

Applying a Unit-Consistent Generalized Matrix Inverse For Stable Control of Robotic Systems

- Mathematics
- 2019

It is well-understood that the robustness of mechanical and robotic control systems depends critically on minimizing sensitivity to arbitrary application-specific details whenever possible. For… Expand

On the Relative Gain Array (RGA) with Singular and Rectangular Matrices

- Mathematics, Computer Science
- Appl. Math. Lett.
- 2019

It is shown that the conventional use of the Moore–Penrose pseudoinverse is inappropriate because it fails to preserve critical properties that can be assumed in the nonsingular case, and it is then shown that such properties can be rigorously preserved using an alternative generalized matrix inverse. Expand

Expression of a Real Matrix as a Difference of a Matrix and its Transpose Inverse.

- Mathematics
- 2019

In this paper we derive a representation of an arbitrary real matrix M as the difference of a real matrix A and the transpose of its inverse. This expression may prove useful for progressing beyond… Expand

Canonical Tensor Scaling

- Mathematics, Computer Science
- ArXiv
- 2020

The canonical positive scaling of rows and columns of a matrix is generalized to the scaling of selected-rank subtensors of an arbitrary tensor for sparse-tensor completion required for generalizations of the recommender system problem beyond a matrix of user-product ratings. Expand

A Method for Online Interpolation of Packet-Loss Blocks in Streaming Video

- Computer Science
- 2019 IEEE Applied Imagery Pattern Recognition Workshop (AIPR)
- 2019

In this paper we examine and apply a linear-time matrix transformation for online interpolation of missing data blocks in frames of a streaming video sequence. We show that the resulting algorithm… Expand

Low-Rank Isomap Algorithm

- Computer Science, Mathematics
- ArXiv
- 2021

The Low-Rank Isomap algorithm is proposed by introducing a projection operator on the embedded graph from the ambient space to a lowrank latent space to facilitate applying the partial eigenvalue decomposition, which leads to reducing the complexity of isomap to a linear order while preserving the structural information during the dimensionality reduction process. Expand

#### References

SHOWING 1-10 OF 23 REFERENCES

A Generalized Matrix Inverse That Is Consistent with Respect to Diagonal Transformations

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2018

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 ...

An efficient computation of generalized inverse of a matrix

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2018

We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per… Expand

Matrix analysis

- Computer Science, Mathematics
- 1985

This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. Expand

Highly Efficient Computation of Generalized Inverse of a Matrix

- Mathematics
- 2016

We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves the 18th order of convergence by using only seven matrix multiplication per… Expand

New methods for computing the Drazin-inverse solution of singular linear systems

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2017

This work presents two new techniques for accelerating the convergence of restarted DGMRES by adding some approximate error vectors or approximate eigenvectors (corresponding to a few of the smallest eigenvalues) to the Krylov subspace. Expand

Generalized inverses: theory and applications

- Mathematics
- 1974

* Glossary of notation * Introduction * Preliminaries * Existence and Construction of Generalized Inverses * Linear Systems and Characterization of Generalized Inverses * Minimal Properties of… Expand

Unit Consistency, Generalized Inverses, and Effective System Design Methods

- Computer Science, Mathematics
- ArXiv
- 2016

This paper examines the potential role of unit consistency as a system design principle. Unit-consistent generalized matrix inverses and unit-invariant matrix decompositions are derived in support of… Expand

Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Constant Coefficients

- Mathematics
- 1976

Let A, B be $n \times n$ matrices, f a vector-valued function. A and B may both be singular. The differential equation $Ax' + Bx = f$ is studied utilizing the theory of the Drazin inverse. A closed… Expand

The relative gain for non-square multivariable systems

- Engineering
- 1990

Abstract Generally speaking, chemical processes in nature are non-square systems with unequal numbers of inputs and outputs. However, only limited tools, e.g. singular-value decomposition (SVD), are… Expand

The Drazin inverse in multibody system dynamics

- Mathematics
- 1993

SummaryThe numerical analysis of multibody system dynamics is based on the equations of motion as differential-algebraic systems. A thorough analysis of the linearized equations and their solution… Expand