On the Cartan decomposition for classical random matrix ensembles
Name
5.0087010.pdf
Description
Published version
Size
6.94 MB
Format
Adobe PDF
Checksum (MD5)
f5359db16643363320a382f0b6c59d21
Author(s)
Edelman, Alan
Jeong, Sungwoo
Date Issued
June 1, 2022
Journal
Journal of Mathematical Physics
Publisher
AIP Publishing
Citation
Edelman, Alan and Jeong, Sungwoo. 2022. "On the Cartan decomposition for classical random matrix ensembles." Journal of Mathematical Physics, 63 (6).
Version
Final published version
Abstract
We complete Dyson’s dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan’s symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra, reveal themselves to be perfectly valid coordinate systems that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decompositions. This, in turn, allows us to produce yet more random matrix theories from a single symmetric space. Yet, again, these random matrix theories arise from matrix factorizations, though ones that we are not aware have appeared in the literature.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1063/5.0087010
