# Eigenvector of infinite matrix

I consider the system of reaction-diffusion PDEs in a ball with Robin boundary condition. It is a Steklov eigenvalue problem (see G Auchmuty (2004) "Steklov eigenproblems and the representation of solutions of elliptic boundary value problems" Numer. Funct. Anal. Optim. 25, 321--348 for details). I have proved that the first eigenvalue has a one-dimesional eigenspace.

On the other hand using the method of separation of variables I want to express the solution in the basis consisting of the Legendre polynomials multiplied by modified Bessel functions of the first kind (see Schiff "Quantum Mechanics" Chapter 4).

From the original PDE system I have obtained an infinite system of linear algebraic equations on expansion coefficients: $(I+A)v=0$, where the matrix $A$ is a band matrix and $\sum A_{ij}^2 <\infty$. From the analysis of regularity of solution to PDE system I know that $v_n \to 0$ faster than $n^{-\alpha}$ for any $\alpha$.

I want to estimate the truncation error, that is: I truncate the system $(I+A)v=0$ to $N$ equations in $N$ variables and find the eigenvector corresponding to the 0 eigenvalue of truncated matrix. Denoting $(I+A)_N$ and $v_N$ the truncated matrix and vector I wold like to ask a question: how the truncation rank $N$ influences the solution - is it possible to prove that $v_N \to v$ in for example $l^2$?

The paper by F Ursell (1996) "Inﬁnite systems of equations. The effect of truncation" Quart. J. Mech. Appl. Math. 49, 217-–233 gives no result because there is an assumption that $\det (I+A) \neq 0$.

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