The generalized characteristic polynomial, corresponding resolvent and their application

We introduced previously the generalized characteristic polynomial defined by \(P_C(\lambda)={\rm det}\,C(\lambda),\) where \(C(\lambda)=C+{\rm diag}\big(\lambda_1,\dots,\lambda_n\big)\) for \(C\in {\rm Mat}(n,\mathbb C)\) and \(\lambda=(\lambda_k)_{k=1}^n\in \mathbb C^n\) and gave the explicit form...

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Bibliographic Details
Published in:arXiv.org 2023-10
Main Author: Kosyak, A V
Format: Article
Language:English
Subjects:
Polynomials
Online Access:Get full text
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Summary:We introduced previously the generalized characteristic polynomial defined by \(P_C(\lambda)={\rm det}\,C(\lambda),\) where \(C(\lambda)=C+{\rm diag}\big(\lambda_1,\dots,\lambda_n\big)\) for \(C\in {\rm Mat}(n,\mathbb C)\) and \(\lambda=(\lambda_k)_{k=1}^n\in \mathbb C^n\) and gave the explicit formula for \(P_C(\lambda)\). In this article we define an analogue of the resolvent \(C(\lambda)^{-1}\), calculate it and the expression \((C(\lambda)^{-1}a,a)\) for \(a\in \mathbb C^n\) explicitly. The obtained formulas and their variants were applied to the proof of the irreducibility of unitary representations of some infinite-dimensional groups.
ISSN:2331-8422