Editing Tridiagonal matrix (section)

==Applications==
The discretization in space of the one-dimensional diffusion or [[heat equation]] 
:<math>\frac{\partial u(t,x)}{\partial t} = \alpha \frac{\partial^2 u(t,x)}{\partial x^2}</math>
using second order central [[finite differences]] results in 

:<math>
\begin{pmatrix}
\frac{\partial u_{1}(t)}{\partial t} \\
\frac{\partial u_{2}(t)}{\partial t} \\
\vdots \\
\frac{\partial u_{N}(t)}{\partial t}
\end{pmatrix}
 = \frac{\alpha}{\Delta x^2} \begin{pmatrix}
-2 & 1 & 0 & \ldots & 0 \\
1 & -2 & 1 & \ddots & \vdots  \\
0 & \ddots & \ddots & \ddots & 0 \\
\vdots  & & 1 & -2 & 1 \\
0 & \ldots & 0 & 1 & -2
\end{pmatrix}
\begin{pmatrix}
u_{1}(t) \\
u_{2}(t) \\
\vdots \\
u_{N}(t) \\
\end{pmatrix}
</math>
with discretization constant  <math>\Delta x</math>. The  matrix is tridiagonal with <math>a_{i}=-2</math> and <math>b_{i}=c_{i}=1</math>. Note: no boundary conditions are used here.