Transformation from integral operator with separable kernel to matrix in eigenvalue problem

TL;DR

This paper transforms the eigenvalue problem of certain integral operators with separable kernels into a matrix eigenproblem, simplifying analysis and computation.

Contribution

It establishes a direct relationship between integral operator eigenvalues/eigenfunctions and matrix eigenpairs, and generalizes eigenfunctions using matrix generalized eigenvectors.

Findings

Eigenvalues of integral operators correspond to matrix eigenvalues.

Eigenfunctions can be derived from matrix eigenvectors and generalized eigenvectors.

The approach simplifies solving Fredholm integral equations of the second kind.

Abstract

This paper investigates the eigenvalue problem of integral operators whose kernels can be expressed as a finite sum of pairwise products of single-variable functions, making them separable. By consdiering the matrix form of the separable kernel in the integral operator, we establish the relationship between the eigenvalues and eigenfunctions of the integral operator and the eigenpairs of a matrix. We next generalize the eigenfunction of an integral operator based on the concept of generalized eigenvectors of matrices, and show that solving the Fredholm integral equation of the second kind reduces to computing matrix eigenpairs and generalized eigenvectors. We also provide several examples to validate our results.