Transformation from integral operator with separable kernel to matrix in eigenvalue problem
Soma Hirai, Ryoto Watanabe, Yuki Nishida, Masashi Iwasaki
TL;DR
The paper addresses the eigenproblem for integral operators with separable kernels by expressing the kernel as $K(x,y)=\sum_{i,j} a_{i,j}p_i(x)q_j(y)$ and showing that the operator's eigenpairs coincide with those of the finite matrix $AB$, with eigenfunctions given by $\phi(x)=\mathbf{p}(x)^T\mathbf{v}$. It extends to generalized eigenfunctions via generalized eigenvectors, enabling biorthogonal expansions $K(x,y)=\sum_{i,j}\phi_i^{(j)}(x)\psi_i^{(j)}(y)$ and deriving operator relations $T_K\phi_i^{(1)}=\lambda_i\phi_i^{(1)}$, $T_K\phi_i^{(j+1)}=\lambda_i\phi_i^{(j+1)}+\phi_i^{(j)}$. The Fredholm equation of the second kind $T_K f = z f + g$ can then be solved by computing matrix eigenpairs and generalized eigenvectors, reducing the problem to solving $(\Sigma - zI)\alpha=(1/z)\beta$. Numerical experiments validate the theory, including cases with distinct and repeated eigenvalues, and demonstrate accurate solution of Fredholm equations under symmetric kernels. The approach offers a finite-precision, matrix-based pathway to spectral analysis and integral equation solutions, particularly advantageous for symmetric separable kernels.
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.