# Rational {1,2}-Pseudo-Inverses: A Flexible and Efficient Solution to Real-Time Matrix Inversion in Multidimensional Systems

### Significance

In multidimensional systems, such as multiple-input multiple-output communication systems and control systems, matrices with rational matrix-valued functions are commonly encountered. These equations arise when modeling and analyzing complex multidimensional systems with spatial and temporal behavior. However, solving these matrix equations can be computationally intensive, leading to limitations in terms of computational time, precision, and energy consumption.

In multiple-input multiple-output communication systems, the decoding of transmitted signals often requires solving large matrix equations. The matrices involved in these equations may dynamically change due to updated parameters or modifications to the operational environment or system set-points. Similarly, in control systems, solving matrix equations is crucial for tasks such as robust stabilization, disturbance rejection, and optimal control. However, the large and evolving nature of these matrix equations poses significant computational challenges. Traditional methods for computing inverses or pseudo-inverses may not be viable due to the required computational resources.

To address these challenges, Dr. Yossi Peretz from Jerusalem College of Technology in Israel, presented the rational {1,2}-pseudo-inverses for rational multivariable matrix-valued functions in a recent study published in the peer-reviewed Journal Mechanical Systems and Signal Processing. The rational {1,2}-pseudo-inverses demonstrated their effectiveness as an alternative solution to real-time matrix inversion problems in various applications, including control systems, image processing, and communication systems.

The study introduced the Moore-Penrose Pseudo-Inverse as a preliminary investigation, which is a powerful mathematical tool extending the concept of matrix inversion to non-square and non-invertible matrices. This pseudo-inverse possesses unique properties and can be computed using methods such as singular value decomposition, QR decomposition, and normal equations.

In linear systems theory, the concept of a pseudo-inverse was explained as a special form of matrix used to solve equations and understand systemic relationships. Under certain conditions, the main theorem stated the existence of a pseudo-inverse for a given matrix. The proof demonstrated that matrices within the system had unique pseudo-inverses with desired properties. Rational {1,2}-pseudo-inverses relaxed the conditions for locating a matrix’s pseudo-inverse. Traditional definitions required satisfying all equations involving the pseudo-inverse, but rational {1,2}-pseudo-inverses considered only the first two equations of the initial conditions. By focusing on these equations, the requirements for rational {1,2}-pseudo-inverses were less stringent compared to conventional pseudo-inverses, providing greater flexibility in the solution-finding process. This relaxation was beneficial when meeting the original conditions was challenging or when a partial solution sufficed. In MIMO communication systems, rational {1,2}-pseudo-inverses enabled efficient decoding and recovery of transmitted signals while considering channel variations, thus improving communication speed and accuracy within the computational and energy limitations of the receiver.

To summarize, Dr. Peretz presented rational {1,2}-pseudo-inverses as an effective solution to real-time matrix inversion problems in multidimensional systems. These pseudo-inverses offer a flexible and efficient approach to solving matrix equations in control systems, image processing, and communication systems. By considering computational limitations and the dynamic nature of the matrices involved, rational {1,2}-pseudo-inverses provide a valuable tool for optimizing computational time, minimizing errors, and optimizing energy consumption in various applications. Indeed, the study’s importance lies in its contribution to overcoming computational limitations, providing flexibility in solving matrix equations, and enabling more efficient operations in multidimensional systems. By introducing rational {1,2}-pseudo-inverses, the study offers valuable insights and practical solutions that can benefit numerous fields where matrix computations are essential.

### Reference

Yossi Peretz. On efficient computation of rational {1,2}-pseudo-inverses for multivariable rational matrix-valued functions and their applications. Mechanical Systems and Signal Processing, Volume 184, January 2023, 109643.