= Solution
At an interior grid point, the five-point <finite difference method> gives
$$
\nabla^2u(ih,jh,t)
\approx\frac1{h^2}
\left(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{ij}\right).
$$
Let
$$
T=
\begin{pmatrix}
-2&1&&\\
1&-2&\ddots&\\
&\ddots&\ddots&1\\
&&1&-2
\end{pmatrix}\in\mathbb R^{m\times m},
$$
where the zero Dirichlet boundary values are already incorporated. Stack the unknowns with the $y$ index varying fastest. Using the <Kronecker product>, define
$$
\boxed{A_x=T\otimes I_m,
\qquad A_y=I_m\otimes T}.
$$
Then the semidiscrete equation is
$$
\boxed{
\frac{d\mathbf u}{dt}
=\frac1{h^2}(A_x+A_y)\mathbf u,
\qquad
\mathbf u(t)\in\mathbb R^{m^2}
}.
$$
Moreover,
$$
A_xA_y
=(T\otimes I)(I\otimes T)
=T\otimes T
=(I\otimes T)(T\otimes I)
=A_yA_x.
$$
\b[Thus the directional matrices commute, as in the <five-point Dirichlet Laplacian as a Kronecker sum>.]
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