Function as an eigenvector for a matrix? So I am currently going through this paper: Link Here
And in section 2.2, it defines $K$ to be a weighted adjacency matrix for a certain rectangular $n$ by $m$ graph, where the weights are all either $1$ or $i$. It then goes on to say that for fixed $j$ and $k$, the function:
$$ f(x,y)=\sin\frac{\pi j x}{m+1}\sin\frac{\pi k y}{n+1}$$
is an eigenvector of $K$. My question is, what does this mean? $K$ is a matrix with strictly $\mathbb{C}$ valued entries, so my understanding is that it can only act on vectors in $\mathbb{C}^n$ for some $n$. What does it mean for this matrix to act on a function?
For example, when $m=2$ and $n=3$, we have the matrix:
$$ K = \begin{bmatrix}
    0 & 0 & 0 & i & 1 & 0 \\
    0 & 0 & 0 & 1 & i & 1 \\
    0 & 0 & 0 & 0 & 1 & i \\
    i & 1 & 0 & 0 & 0 & 0 \\
    1 & i & 1 & 0 & 0 & 0 \\
    0 & 1 & i & 0 & 0 & 0 \\
\end{bmatrix}$$
In this case, what are the eigenvectors that it is referring to?

It appears that the standard basis of the vector space the linear operator $K$ acts on is most easily enumerated by two indexes $x,y$ where $x\in\{1,\ldots,m\}$ and $y\in\{1,\ldots,n\}$. To specify a vector in this space, we need to specify the component this vector has for each possible combination of $x$ and $y$ from these sets. That is, to specify a vector, we may specify a function $f\colon\{1,\ldots,m\}\times\{1,\ldots,n\}\to\Bbb C$. And that is precisely what happens here.
