A Pertubation Matrix and Its Eigen Functions

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DOI: https://doi.org/10.28924/2291-8639-23-2025-313
Published: Nov 28, 2025
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M. Surya Priya, N. Nathiya, A Pertubation Matrix and Its Eigen Functions, Int. J. Anal. Appl., 23 (2025), 313.

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M. Surya Priya, N. Nathiya

Abstract

On a finite graph \(\mathcal{N}\) with a set of possibly non-symmetric transition indices \(\{c(a,b)\}\), \(c(a,b)\geq 0\), \(c(a)=\sum\limits_{b}c(a,b)\leq 1\), an operator \(Ku(a)=(I-A)u(a)=u(a)-\sum c(a,b)u(b) \) is defined. We discuss properties of the operator \(K\). We prove that for an eigen function \(\xi (a) \) with positive entries, \(K\xi (a)=\rho \xi (a)\) where \(\rho >0\) and show that the eigen value \(\rho\) is the smallest in the following sense: if for an eigen function \(\eta (a)\), \(K \eta (a)=\beta \eta (a) \) then \(Re \beta >\rho\). This result establishes the uniqueness and minimality of the positive eigenvalue associated with the positive eigenfunction. Finally, it is proven that the set \(\mathfrak{F}=\{u:Ku(a)\geq 0\}\) forms a convex cone that is a lattice under the natural order.

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