Python Sparse matrix inverse and laplacian calculation

Question

I have two sparse matrix A (affinity matrix) and D (Diagonal matrix) with dimension 100000*100000. I have to compute the Laplacian matrix L = D^(-1/2)*A*D^(-1/2). I am using scipy CSR format for sparse matrix.

I didnt find any method to find inverse of sparse matrix. How to find L and inverse of sparse matrix? Also suggest that is it efficient to do so by using python or shall i call matlab function for calculating L?

Answer 1

In general the inverse of a sparse matrix is not sparse which is why you won't find sparse matrix inverters in linear algebra libraries. Since D is diagonal, D^(-1/2) is trivial and the Laplacian matrix calculation is thus trivial to write down. L has the same sparsity pattern as A but each value A_{ij} is multiplied by (D_i*D_j)^{-1/2}.

Regarding the issue of the inverse, the standard approach is always to avoid calculating the inverse itself. Instead of calculating L^-1, repeatedly solve Lx=b for the unknown x. All good matrix solvers will allow you to decompose L which is expensive and then back-substitute (which is cheap) repeatedly for each value of b.