# Inverse of symmetric matrix that is almost diagonal

I have an $$N\times N$$ matrix $$\mathbb X$$ with entries:

$$X_{ij} = x_i\delta_{ij} + y_i(\delta_{i+m,j}+\delta_{i,j+m})$$

where $$1 \le m \le N$$, and $$x_i,y_i$$ are given numbers.

Is there an analytical formula for the inverse of $$\mathbb X$$? I've found numerically that $$\mathbb X^{-1}$$ is also sparse, with non-zeros on the main diagonal and a few other diagonals.

Please note that your matrix $$X$$ is actually the result of the row and column permutation of a block diagonal matrix with $$m$$ tridiagonal blocks. You can permute the rows and columns such that the original rows/columns $$1,1+m,1+2m,\ldots,1+\left\lfloor\frac{N-1}{m}\right\rfloor m,\\ 2,2+m,2+2m,\ldots,2+\left\lfloor\frac{N-2}{m}\right\rfloor m,\\ \vdots \\ m,m+m,m+2m,\ldots,m+\left\lfloor\frac{N-m}{m}\right\rfloor m$$ form the new rows/columns $$1,\ldots,N.$$ You will get some tridiagonal blocks of size $$\left\lfloor\frac Nm\right\rfloor$$ and some tridiagonal blocks of size $$\left\lceil\frac Nm\right\rceil,$$ which you can invert in isolation using the formula given in this link about tridiagonal matrices. In the end, you only have to undo the permutations.