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?

6$\begingroup$ Infinite matrices don't have eigenvalues in general. This is a subtle matter and "infinite matrix" is not the correct keyword for understanding it: see en.wikipedia.org/wiki/… . $\endgroup$ – Qiaochu Yuan Oct 13 '12 at 23:44

8$\begingroup$ So you want to learn about the spectral theory of operators on Hilbert space? The basic general principles and existence theorems will be covered in standard books such as those of JB Conway or W Rudin (or RG Douglas's book). However, finding the spectrum of nonnormal operators is in general a very hard business, so I think you would be well advised to think about the particular structure of the "infinite matrices" that you are interested in. $\endgroup$ – Yemon Choi Oct 14 '12 at 0:42

5$\begingroup$ Infinite matrices represent linear operators between spaces of sequences. But it should be clear that not any infinite matrice represent an linear operator, and not any linear operator has an infinite matrix representation. A nice book that addresses these and other foundational issues is HalmosSunder's Bounded Integral operators on $L^2$ spaces. projecteuclid.org/… $\endgroup$ – Pietro Majer Oct 14 '12 at 11:21

4$\begingroup$ Perhaps a good place to start is to check out the history of infinite matrices (including various discussions around eigenvalues): springerlink.com/content/jw45823632u8qj65 $\endgroup$ – Benjamin Dickman Oct 14 '12 at 11:49

8$\begingroup$ You say below "Please do not advise me to read the general theory of linear operator in Hilbert space, seriously I know this stuff. But I just don't know how should deal with infinite matrices." I find this attitude hard to understand. If you know "this stuff" then you should know that the operator might have nonempty residual spectrum, might have approximate eigenvalues that are mot eigenvalues, and so on. Per's answer only applies to Toeplitz matrices. You cannot just regard the spectra of infinite matrices as being limits of eigenvalues of naive truncations $\endgroup$ – Yemon Choi Oct 14 '12 at 18:59
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 ...

$\begingroup$ Excellent example! This is indeed what I am worried about. Thanks a lot! $\endgroup$ – Youzhou Zhou Oct 15 '12 at 14:45
For me, infinite matrix means an operator on $\ell^2(\mathbb N)$ (or sometimes $\ell^2(\mathbb Z)$, but usually referred to biinfinite 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.
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.
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

1$\begingroup$ Thank you so much! So we truncate the infinite matrix and find the eigenvalues, then we take limits. If the limits exist, then we regard the limit as the eigenvalue of infinite matrices. Do you think it is a legitimate treatment of eigenvalues of infinite matrices? Please do not advise me to read the general theory of linear operator in Hilbert space, seriously I know those stuff. But I just don't know how should we deal with infinite matrices. Do you think infinite sparse matrices are easier to deal with? Thank you so much! $\endgroup$ – Youzhou Zhou Oct 14 '12 at 16:56

3$\begingroup$ Hm, I would not dare to say that this in general is valid; I guess it depends on the application. Intuitively, if taking a kxktruncated matrix and the eigenvalues are "close" to what you'd expect, or there is nice convergence as k grows, then it is worth studying. But you can certainly construct a series of truncated (nonToeplitz) matrices such that the series of eigenvalues do not converge. I have not dealt with sparse matrices. $\endgroup$ – Per Alexandersson Oct 14 '12 at 18:15