Is there an analytical way or a good approximation or any other mathematical method to diagonalise a sparse (symmetric) matrix with elements only onsome diagonals?
For example $$ \begin{bmatrix} B & 0 & 0 & A & 0\\ 0 & B & 0 & 0 & A\\ 0 & 0 & B & 0 & 0\\ A & 0 & 0 & B & 0\\ 0 & A & 0 & 0 & B \\ \end{bmatrix} $$
or similar...
(is there an index notation way of writing the above matrix? Like $A_{m,n} = \cdots$?)
By using $Xv_i= \lambda_i v_i$ you can derive the eigenvectors: $(\frac{1}{\sqrt{2}},0,0,\pm\frac{1}{\sqrt{2}},0)$, $(0,0,1,0,0)$, $(0, \frac{1}{\sqrt{2}},0,0,\pm\frac{1}{\sqrt{2}})$, with corresponding eigenvalues $\lambda =(B\pm A, B, B\pm A)$. Let $Q$'s columns be made up of the eigenvectors (in the given order), then: $X= Qdiag(\lambda)Q^T$, where $X$ is your initial matrix. Note that this can be trivially extended to higher dimensions for a matrix with the same structure.