Our teacher told us that the following matrix is definite positive, how to prove it?
Note: this is not an excercise, I simply do not like to believe one's words without a proof.
$$ \begin{pmatrix} 2 & -1 & 0 & \ldots & 0 \\ -1 & 2 & -1 & \ldots & 0 \\ 0 & -1 & \ddots & \ddots &\vdots \\ \vdots & \vdots & \ddots & 2 & -1 \\ 0 & 0 &\ldots & -1 & 2 \end{pmatrix} $$
Curiosity: this matrix comes out when trying to solve numerically the Poisson equation in one dimension.