Fifty Three Matrix Factorizations: A Systematic Approach
Name
21m1416035.pdf
Description
Published version
Size
26.49 MB
Format
Adobe PDF
Checksum (MD5)
2ff6c1d702259661669fd8db1a305bdb
Author(s)
Edelman, Alan
Jeong, Sungwoo
Date Issued
June 2023
Journal
SIAM Journal on Matrix Analysis and Applications
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
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 the 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 the 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.
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DOI of Published Version
https://doi.org/10.1137/21M1416035
