Consider $A = \mbox{Mat}(\mathbb{R}_{K \times K} )$, where $K = (N-1)(M-1)$. This matrix has tridiagonal part: $A_{i,i} = \frac{2}{a^2} + \frac{2}{b^2}$, $A_{i,i+1} = -\frac{1}{a^2}$, $A_{i,i-1} = -\frac{1}{a^2}$, and there is two other non-zero diagonals: $A_{i,i+M} = -\frac{1}{b^2}$, $A_{i,i-M} = -\frac{1}{b^2}$. What can we say about its eigenvalues, about inverse matrix of this one? Any hints.
Example: let $N = M = 4$, hence $a = b = \frac{1}{4}$, $K = 9$ and we have:
$$\begin{pmatrix} 36 & -9 & 0 & 0 & -9 & 0 & 0 & 0 & 0\\ -9 & 36 & -9 & 0 & 0 & -9 & 0 & 0 & 0\\ 0 & -9 & 36 & -9 & 0 &0 &-9 & 0 & 0 \\ 0 & 0 & -9 & 36 & -9 & 0 &0 &-9 & 0\\ -9 & 0 & 0 & -9 & 36 & -9 & 0& 0& -9 \\ 0 & -9 & 0 & 0 &-9 & 36 &-9 & 0& 0 \\ 0 &0&-9& 0 &0&-9&36&-9&0\\ 0 & 0 & 0 & -9 & 0 & 0& -9&36&-9\\ 0 & 0&0&0&-9&0&0&-9&36 \end{pmatrix}$$
This problem occurs because of difference scheme according to Stokes equation.
$N.B.$ Actually this question become more pragmatic and I've interested in some algorithm for approximation of inverse matrix. So I have Toeplitz-matrix with size $(2\cdot N\cdot M) \times (2\cdot N\cdot M)$ does anybody knows the algorithm of finding inverse matrix for such matrices??