Fifty Three Matrix Factorizations: A systematic approach (2104.08669v5)

Abstract: The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most of which we believe to be new. Our systematic approach, inspired by the generalized Cartan decomposition of Lie theory, also encompasses known factorizations such as the SVD, the symmetric eigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factorization, and others thereby covering familiar matrix factorizations as well as ones that were waiting to be discovered. We suggest that Lie theory has one way or another been lurking hidden in the foundations of the very successful field of matrix computations with applications routinely used in so many areas of computation. In this paper, we investigate consequences of the Cartan decomposition and the little known generalized Cartan decomposition for matrix factorizations. We believe that these factorizations once properly identified can lead to further work on algorithmic computations and applications.

• The paper presents a systematic framework that unifies 53 matrix factorizations by leveraging generalized Cartan and Lie group decompositions.
• The paper details algorithmic derivations of established decompositions like SVD and CSD through iterative exploitation of Lie group symmetries.
• The paper demonstrates implications for numerical linear algebra and quantum information, paving the way for innovative computational techniques.

A Systematic Catalog of Matrix Factorizations Inspired by Lie Theory

In the given paper, Alan Edelman and Sungwoo Jeong deliver a comprehensive exploration into the field of matrix factorizations, offering an extensive catalog of 53 different kinds based on a systematic approach underpinned by Lie theory, particularly leveraging the generalized Cartan decomposition. This research constitutes an attempt at unifying various matrix factorizations within a structured framework, establishing connections previously underexplored, and thereby potentially enlightening new computational techniques and applications across disciplines.

Overview of Matrix Factorizations

Matrix factorization sits prominently within linear algebra as a cardinal concept, extending its utility from theoretical underpinnings to practical implementations in fields such as physics, engineering, and data science. This work notably embraces both classical and modern mathematical methodologies to systematically derive these factorizations. Central to this is the employment of notions from Lie groups and algebras, the Cartan decomposition, and its generalization, the KAK and KAK$^{*}$ decompositions.

The paper delineates the derivation of each factorization by initially considering a suitable classical Lie group and then executing a successive breakdown using the algebraic structure intrinsic to the group's symmetries. Through such a lens, the researchers revisit and unify known decompositions like the Singular Value Decomposition (SVD) and the Cartan-based CS Decomposition (CSD) while also revealing what they posit as new factorizations within the same structural vein.

Technical Foundations

The backbone of this exposition lies in theoretical constructs such as the Cartan decomposition for symmetric spaces, alongside the so-called generalized Cartan decomposition pertinent for pseudo-Riemannian manifolds. The research elaborates on algorithmically arriving at factorizations through careful selection of Lie groups, implementation of appropriate involutions on associated Lie algebras, and subsequent segmentation into eigenspace components to produce the novel analytical decompositions cataloged.

The paper journeys through various constructs, including matrix factorizations pertaining to orthogonal/unitary matrices (e.g., ODO decomposition, QDQ decomposition), those adaptively illustrated through the context of invertible matrices (e.g., hyperbolic SVD), to more sophisticated constructs involving symplectic matrices and symplectic structured SVDs.

Implications and Prospects

While providing strong numerical results, including positive diagonal forms and various orthogonal/unitary components, these results hold profound implications for fields that leverage large-scale computations and symbolic processing. These include quantum information processing, which can benefit from insights on canonical forms, as well as advancements in numerical linear algebra techniques.

The theoretical implications also run deep, contributing to a broader understanding of matrix spaces as they interact with Lie group symmetries — advancing beyond traditional boundaries in harmonic analysis and representation theory. Looking ahead, these foundations could spur future developments by encouraging algorithmic exploration rooted in symmetries and decompositions informed by both compact and noncompact forms of these algebraic objects.

Conclusion

This paper positions itself as a staple reference for researchers in numerical linear algebra, applied mathematics, and any domain where the profound intricacies of matrix factorizations intersect with computational needs. By setting up a systematized framework under the influence of Lie theory and demonstrating its potential through various introduced and reconciled factorizations, it invites ongoing dialogue and exploration into deriving efficient computational schemas influenced by elegant mathematical structures.