BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alan Edelman (MIT) DTSTART:20210610T130000Z DTEND:20210610T140000Z DTSTAMP:20260404T102103Z UID:OxfordRMT/26 DESCRIPTION:Title: 53 Matrix Factorizations\, generalized Cartan\, and Random Matrix Theor y\nby Alan Edelman (MIT) as part of Oxford Random Matrix Theory Semina rs\n\n\nAbstract\nAn insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalue s or linearity\, it would be “matrix factorizations.” We will discuss a blueprint to generate 53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927)\, Harish-Chandra (1956)\, and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wi gner (1968) discovered factorizations such as the svd in passing in a way that was buried and only eight authors have referenced that work. Ironical ly Wigner referenced Sigurður Helgason (1962) but Wigner did not recogniz e his results in Helgason's book. This work also extends upon and complete s open problems posed by Mackey²&Tisseur (2003/2005).\n\nClassical result s of Random Matrix Theory concern exact formulas from the Hermite\, Laguer re\, Jacobi\, and Circular distributions. Following an insight from Freema n Dyson (1970)\, Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensemb les. We present a completed theory based on the generalized Cartan distrib ution. Furthermore\, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampl ing algorithms and the derivation of the joint probability density of A.\n \nJoint work with Sungwoo Jeong.\n LOCATION:https://researchseminars.org/talk/OxfordRMT/26/ END:VEVENT END:VCALENDAR