Method of Lines Diffusion Problem 
Consider the method of lines applied to the diffusion equation
  in one space dimension, $u_t = au_{xx}$, with $a > 0$, $a$ constant, $u = 0$ at $x =0$, $x = 1$ for $t ≥ 0$, and with given initial values.  Formulate the method
  of lines using the central difference approximation to the derivative
  $u_{xx}$. To arrive at a linear constant coefficient ODE system $y' = Ay$ with $A$ symmetric.
  Find the eigenvalues of $A$ to determine whether the problem is unstable,
  stable or asymptotically stable.

So this is what I have so far..
$u_t=u_{xx}$ with $u(0,t)=0$ and $u(1,t)=0$ as the conditions.
So by method of lines and centeral difference approximation...bounded (0,1) thus $0<i<N$
$\frac{du(t)}{dt} =a \frac{u_{x_{i+1}}-2u_{x_i}-u_{_{x_i-1}}}{\delta x^2}$
Now using the conditions and incrementing because we can't have $x_i=-1$ for the first starting point of $u_{i-1}=u_{0-1}=u_{-1}$...we will have the new bounds be $1<i<N+1$
$\frac{du(t)}{dt} =a \frac{u_{x_{2}}-2u_{x_1}-u_{_{x_0}}}{\delta x^2}$
which reduces with the $u_{x(0)}=0$ and $u_{x(1)}=0$ by the conditions given...
$\frac{du(t)}{dt} =a \frac{u_{x_{2}}-0-0}{\delta x^2}$
$\frac{du(t)}{dt} =a \frac{u_{x_{2}}}{\delta x^2}$
Now I'm not 100% sure if I did the method of lines correct... mostly with the reindexing of the bound...
Now I also don't understand what my next step would be in terms of solved for the A matrix.
All I can think of is...but it gets me nowhere.
$\frac{du(t)}{dt} -a \frac{u_{x_{2}}}{\delta x^2} = 0$
Any help would be greatly appreciated. Thanks.
 A: The space $x\in [0,1]$ is discretized using a regular grid with abscissas $x_i = i\, \Delta x$, where $\Delta x = 1/N_x$ and $i=0, \dots, N_x$. Using a second-order centered difference for $u_{xx}$, the interior grid node values $u_i(t)$ satisfy $u'_i(t) = a \left(u_{i+1}(t) - 2u_{i}(t) + u_{i-1}(t)\right)/{\Delta x}^2$ for $i=1, \dots, N_x-1$. At the boundaries of the domain, we have $u_0(t) = 0$ and $u_{N_x}(t) = 0$. Therefore, we get the following system of differential equations satisfied by the unknown grid node values:
$$
\underbrace{
\left(\begin{array}{c}
u'_1(t) \\
u'_2(t) \\
\vdots \\
u'_{N_x-2}(t) \\
u'_{N_x-1}(t)
\end{array}
\right)}_{y'(t)}
=
\underbrace{\frac{a}{{\Delta x}^2}
\left(\begin{array}{ccccc}
-2 & 1 & 0 &  & \dots \\
1 & -2 & 1 & 0 & \dots \\
 & \ddots & \ddots & \ddots & \\
\dots & 0 & 1 & -2 & 1 \\
\dots & & 0 & 1 & -2
\end{array}\right)}_{A}
\;
\underbrace{\left(\begin{array}{c}
u_1(t) \\
u_2(t) \\
\vdots \\
u_{N_x-2}(t) \\
u_{N_x-1}(t)
\end{array}
\right)}_{y(t)} .
$$
It remains to compute the spectrum of $A$.
