The Moore-Penrose Pseudoinverse. A Tutorial Review of the Theory

TL;DR

This paper provides a comprehensive tutorial review of the Moore-Penrose pseudoinverse, covering its theory, applications, and computational methods, aimed at both practitioners and researchers in science and mathematics.

Contribution

It offers a complete, self-contained tutorial on the Moore-Penrose pseudoinverse, including definitions, theory, spectral theorem, and simple algorithms for computation.

Findings

Clarifies the theoretical foundations of the pseudoinverse.

Provides an accessible algorithmic approach for computation.

Highlights applications across various scientific fields.

Abstract

In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore-Penrose pseudoinverse and in related sub jects, like the singular values decomposition theorem for matrices. In this note we present a tutorial review of the theory of the Moore-Penrose pseudoinverse. We present the first definitions and some motivations and, after obtaining some basic results, we center our discussion on the Spectral Theorem and present an algorithmically simple expression for the computation of the Moore-Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented and for those who already have some working knowledge of the sub ject.