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SUMMARY:Rongwei Yang (University at Albany\, SUNY)
DTSTART:20241029T060000Z
DTEND:20241029T070000Z
DTSTAMP:20260423T021222Z
UID:OISTRTS/62
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OISTRTS/62/"
 >Linear algebra in several variables</a>\nby Rongwei Yang (University at A
 lbany\, SUNY) as part of OIST representation theory seminar\n\n\nAbstract\
 nMany mathematical and scientific problems concern systems of linear opera
 tors $(A_1\, ...\, A_n)$. Spectral theory is expected to provide a mechani
 sm for studying their properties\, just like the case for an individual op
 erator. However\, defining a spectrum for non-commuting operator systems h
 as been a difficult task. The challenge stems from an inherent problem in 
 finite dimension: is there an analogue of eigenvalues in several variables
 ? Or equivalently\, is there a suitable notion of joint characteristic pol
 ynomial for multiple matrices $A_1\, ...\, A_n$? A positive answer to this
  question seems to have emerged in recent years.\n\n<b>Definition</b>. Giv
 en square matrices $A_1\, ...\, A_n$ of equal size\, their characteristic 
 polynomial is defined as \n\\[Q_A(z):=\\det(z_0I+z_1A_1+\\cdots+z_nA_n)\, 
 z=(z_0\, ...\, z_n)\\in \\mathbb{C}^{n+1}.\\] Hence\, a multivariable anal
 ogue of the set of eigenvalues is the <i>eigensurface</i> (or <i>eigenvari
 ety</i>) \n $Z(Q_A):=\\{z\\in \\mathbb{C}^{n+1}\\mid Q_A(z)=0\\}$. This ta
 lk will review some applications of this idea to problems involving projec
 tion matrices and finite dimensional complex algebras. The talk is self-co
 ntained and friendly to graduate students.\n
LOCATION:https://researchseminars.org/talk/OISTRTS/62/
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