Eigenvalues of infinite matrices - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T04:29:39Z http://mathoverflow.net/feeds/question/109567 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices Eigenvalues of infinite matrices John Young 2012-10-13T23:36:18Z 2012-10-15T11:34:12Z <p>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?</p> http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109573#109573 Answer by Delio Mugnolo for Eigenvalues of infinite matrices Delio Mugnolo 2012-10-14T00:54:55Z 2012-10-14T00:54:55Z <p>to present an example: if the considered matrix is the adjacency matrix of a graph, there are relatively involved graph theoretical criteria <em>just</em> 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.</p> http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109591#109591 Answer by Adrien Hardy for Eigenvalues of infinite matrices Adrien Hardy 2012-10-14T09:22:50Z 2012-10-14T09:22:50Z <p>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. </p> http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109602#109602 Answer by Per Alexandersson for Eigenvalues of infinite matrices Per Alexandersson 2012-10-14T11:32:59Z 2012-10-14T11:32:59Z <p>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: <a href="http://arxiv.org/abs/1208.5607" rel="nofollow">http://arxiv.org/abs/1208.5607</a></p> http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109663#109663 Answer by Gottfried Helms for Eigenvalues of infinite matrices Gottfried Helms 2012-10-15T00:48:40Z 2012-10-15T11:34:12Z <p>One simple example with a special matrix, which has somehow "a continuum" as eigenvalue...<br> 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 $$\small \begin{bmatrix} K &amp; . &amp; . &amp; . &amp; \cdots \\ a &amp; K &amp; . &amp; . &amp; \cdots \\ b &amp; a &amp; K &amp; . &amp; \cdots \\ c &amp; b &amp; a &amp; K &amp; \cdots \\ \vdots &amp; \vdots &amp; \vdots&amp; \vdots &amp; \ddots \end{bmatrix}$$ From the properties of finite matrices we would expect, that <em>K</em> is an eigenvalue. But consider a type of an infinite vector</p> <p>$$V(x) = [1,x,x^2,x^3,x^4,\ldots ]$$ 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 <strong><em>F</em></strong> 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 <strong><em>F</em></strong> contradicting the "naive" extrapolation from the finite truncation of the matrix ...</p>