Eigenvalues of infinite matrices - MathOverflow [closed]

Eigenvalues of infinite matrices

2012-10-13T23:36:18Z

I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can someone tell me how to determine the spectrum of infinite matrices?

Answer by Delio Mugnolo for Eigenvalues of infinite matrices

2012-10-14T00:54:55Z

to present an example: if the considered matrix is the adjacency matrix of a graph, there are relatively involved graph theoretical criteria just to decide whether the spectral radius is an eigenvalue, let alone further spectral values. i think you should make yourself clearer what exactly you would like to know.

Answer by Adrien Hardy for Eigenvalues of infinite matrices

2012-10-14T09:22:50Z

For me, infinite matrix means an operator on $\ell^2(\mathbb N)$ (or sometimes $\ell^2(\mathbb Z)$, but usually referred to bi-infinite matrix). Concerning the eigenvalues, you thus may just look at the general theory concerning operator on Hilbert spaces, as already pointed out in the comments above.

Answer by Per Alexandersson for Eigenvalues of infinite matrices

2012-10-14T11:32:59Z

I have done some research on banded Toeplitz matrices, (where we consider a sequence of finite matrices, and show where the eigenvalues accumulate). There is also quite old literature on this subject, see references in this paper I shamelessly advertise: http://arxiv.org/abs/1208.5607

Answer by Gottfried Helms for Eigenvalues of infinite matrices

2012-10-15T00:48:40Z

One simple example with a special matrix, which has somehow "a continuum" as eigenvalue...
Consider some function $ f(x) = K + ax + bx^2 + cx^3 + ... $ having a nonzero radius of convergence. Then think of the infinite matrix of the form $$ \small \begin{bmatrix} K & . & . & . & \cdots \\ a & K & . & . & \cdots \\ b & a & K & . & \cdots \\ c & b & a & K & \cdots \\ \vdots & \vdots & \vdots& \vdots & \ddots \end{bmatrix} $$ From the properties of finite matrices we would expect, that K is an eigenvalue. But consider a type of an infinite vector

$$ V(x) = [1,x,x^2,x^3,x^4,\ldots ] $$ with a scalar parameter $x$ from the range of convergence, then $$ V(x) \cdot F = f(x) \cdot V(x) $$ This means also: any vector $V(x)$ is an eigenvector of the matrix F and corresponds to the eigenvalue $f(x)$. If now $f(x)$ is entire, for instance the exponential function $ f(x)=\exp(x)$, then any value from the complex plane (except $0$ because $\exp(x)$ is never $0$) "is an eigenvalue" of F contradicting the "naive" extrapolation from the finite truncation of the matrix ...