Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse

Core Concepts

The core message of this article is to provide a constructive characterization of matrices satisfying the reverse order law for the Moore-Penrose pseudoinverse. The authors show that any matrix satisfying this law can be obtained through a specific construction involving the right singular vectors of the given matrix.

Abstract

The article focuses on characterizing matrices that satisfy the reverse order law for the Moore-Penrose pseudoinverse, i.e., (AB)+ = B+A+. The key highlights and insights are:

The authors provide a sufficient condition (Theorem 3.1) and a related necessary condition (Theorem 5.2) that allow for the explicit construction of all matrices B satisfying the reverse order law for a fixed matrix A.

The construction is done in terms of the singular value decompositions (SVDs) of matrices A and B. Specifically, the condition involves the right singular vectors of A and the left singular vectors of B.

The authors show that the reverse order law holds if and only if the column spaces of A* and B are related in a specific way, as described by the equivalent conditions in Theorem 1.2.

The authors also provide a geometric interpretation of the conditions for the reverse order law in terms of the principal angles between the column spaces of A* and B (Theorem 8.2).

The authors establish the similarity between the equivalent conditions for B+A+ being an {1, 2}-inverse of AB and being an {1, 2, 3, 4}-inverse of AB (Tables 1 and 2).

The authors parameterize all possible SVD decompositions of a fixed matrix and give Greville-like equivalent conditions for B+A+ being a {1, 2}-inverse of AB with a geometric insight in terms of the principal angles between the column spaces of A* and B.

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Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse

by Oska... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02843.pdf

Deeper Inquiries

How can the results in this article be extended to characterize matrices satisfying the reverse order law for other types of generalized inverses, beyond the Moore-Penrose pseudoinverse

The results in this article can be extended to characterize matrices satisfying the reverse order law for other types of generalized inverses by considering the properties and relationships between the matrices involved. For example, for the Drazin inverse or the group inverse, similar conditions related to the singular value decompositions of the matrices could be explored. By analyzing the specific properties of these generalized inverses and their interactions with the original matrices, it may be possible to establish equivalent conditions for the reverse order law similar to those derived for the Moore-Penrose pseudoinverse.

What are the potential applications of the characterization of matrices satisfying the reverse order law for the Moore-Penrose pseudoinverse in areas such as optimization, control theory, or numerical linear algebra

The characterization of matrices satisfying the reverse order law for the Moore-Penrose pseudoinverse has various potential applications in different fields. In optimization, understanding the properties of matrices that satisfy this law can lead to more efficient algorithms for solving optimization problems involving matrix operations. In control theory, these characterizations can be utilized to design robust control systems with improved stability and performance. In numerical linear algebra, the insights gained from these characterizations can enhance the development of numerical methods for solving linear systems and eigenvalue problems, leading to more accurate and reliable computations.

Can the geometric interpretation of the conditions for the reverse order law in terms of principal angles be further developed to provide additional insights or connections to other areas of mathematics

The geometric interpretation of the conditions for the reverse order law in terms of principal angles can be further developed to provide deeper insights and connections to other areas of mathematics. By exploring the geometric relationships between the subspaces spanned by the columns of the matrices involved, one can potentially uncover new geometric properties and structures related to matrix operations and inverses. This geometric perspective may also offer connections to topics such as manifold learning, subspace clustering, and geometric optimization, providing a richer understanding of the underlying mathematical principles and their applications.