I am interested in calculating the eigenvalues of integral kernels, but I figured it would be useful to get some intuition about the simplest one first: the identity operator.
The interval $[-L,L]$ has a countable set of basis functions, so there is a sense in which the identity operator may be written as a matrix \begin{align} I_n = \left[\begin{array}{cccc}1&0&0&\cdots\\0&1&0\\0&0&1\\\vdots&&&\ddots\end{array}\right] \end{align} where $I_n$ is the $n\times n$ identity matrix, and the idenity on the interval $[-L,L]$ is $\lim_{n\to\infty} I_n$. As we take the limit $L\to\infty$, the set of basis functions becomes uncountable, and the identity operator becomes a Dirac $\delta$-function. Is there a sense in which the Dirac $\delta$ may be thought of as a limit of matrices? Given the transition from countable to uncountable, I suspect not, but perhaps I am mistaken.
In any case, the matrix representation of the countable basis is nice because it provides an easy-to-understand sense in which I may identify/calculate the eigenvaleus of an operator. If the uncountable case may not be thought of as a limit of the countable case, how should I think about calculating the eigenvalues of an integral operator?