Intuition behind: Integral operator as generalization of matrix multiplication So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications.
In particular, if:
$$Lf(x) = \int_{X} k(x,y) f(y) du(y)$$
So then, naturally, if $\mathbf{A}$ is an $m,n$ matrix with entries $a_{ij}$ and $\mathbf{u} \in \mathcal{C}^{n}$, $\mathbf{A}\mathbf{u} \in \mathcal{C}^{m}$, we have:
$(\mathbf{A}\mathbf{u})_{i} = \sum_{j=1}^{n} a_{ij} u_{j}, i=1 \ldots m$
So I always see something similar to the following statement:
The entries $k(x,y)$ are analogous to the entries $a_{ij}$ of matrix $\mathbf{A}$ and the values $Lf(x)$ are analogous to the entries $(\mathbf{A}\mathbf{u})_{i}$
I have never seen any example or more detail regarding this statement. Can somebody make this a bit clearer?

Here is my thought, the $x$ defines the 'row', while the $y$ defines the column. 
So then if we wanted to, we could define a sequence $\{x_{0}, x_{1}, \ldots\}$ to then 'sample' the output of the operator integral $L$. For instance, if $L$ mapped $f$ to a certain region $P$, we would want to define our sequence to exist only in this region $P$ in order to save computation. If in this region, the functions $Lf$ had significant norm/power only in a subspace or section of this region $P_{k}$, we could 'sample' this subspace only -- ie $\{x_{0}, x_{1}, \ldots\} \in P_{k}$ and still get a 'reasonable' approximation to the resulting function/signal $Lf$.
Similarly, we could 'sample' $f$ on its domain with  $\{y_{0}, y_{1}, \ldots\}$ and reduce the computation even further, when the sequence captures 'most of the information' about $f$ that is.

More thoughts about this. Matrix $A$ can be thought of as a linear operator from $\mathbb{R}^n$ to $\mathbb{R}^n$. In a similar way your integral transform $L$ is an operators from a (Hilbert) space of functions to a different space.
Just like you can define characteristics of $A$ (like eigenvalues and eigenvectors), and talk about basis of its image, so too you can do the same to $L$.
For an in-depth example of continuous and discrete transformations with similar eigenvalues and "related" eigenvectors, look at continuous and discrete Fourier transforms.
