Having read: What are eigenvalues of higher order finite differences matrices? I am still unclear how you would do this for an arbitrary matrix: $$A=\begin{pmatrix} a&b&e&0&0&0&0\\ d&a&b&e&0&0&0\\ c&d&a&b&e&0&0\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ 0&0&c&d&a&b&e\\ 0&0&0&c&d&a&b\\ 0&0&0&0&c&d&a \end{pmatrix}\in\mathbb{R}^{n\times n}$$ Especially if there are boundary conditions imposed on your system. I have seen an example pop up on wikipedia over the summer which also uses Chebyshev polynomials https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative but there's no reference or explanation where the Chebyshev polynomials come from.
For example, lets take the original problem to be $$ u_{t}=Du_{xx}-c u_{x}$$ where $D,c$ are constants, and Boundary conditions: $$ u(t,0)=0,\quad u_{x}(t,L)=0 $$ and some well defined initial condition, $u(0,x)=...$ and there's some n-diagonal matrix, as above, representing some finite difference scheme on a uniform grid on $[0,1]$. So that you get some ODE system $$u_{t} = Au + b$$ $b$ being the boundary data, (I realize that the first and last rows should account for the BC's). So it's respective eigenvalue problem should be something like: $$(A-\lambda I)v=0$$
Can someone explain where the Chebyshev polynomials come from and how to apply the resulting recursion to them? Is there an alternative method to compute the $k$-th Eigenvalue?
