I want to know that what is the efficient way to inverse a sparse matrix.
I want to implement this type of equation using sparse matrix.
$ \Sigma\ = \Omega^{-1}$ Here $ \Omega $ is a Sparse Matrix or it is efficient to say that $ \Omega $ is a band diagonal matrix whose diagonal and off diagonal element have some values and rest of the entries are zero.
My equation is
$ \mu = \Omega^{-1} \ Xi $
Now what is the efficient way to compute Matrix inverse?
In general there is no efficient way to invert a sparse matrix, since the inverse of a sparse matrix need not be sparse. But you have much more structure because your matrix is band diagonal. Moreover, based on my reading of your question, you only have a single off-diagonal band. In this case you don't need to actually an inverse of $\Omega$. You can just do backward substitution on $\Omega$ and it will be efficient.
See examples in https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/lu.pdf if you need more help.