[petsc-users] Best way to compute the null space of a sparse maximum-rank rectangular matrix

[Luca Argenti]

> Yes, that is true. Maybe it is possible to add an option to workaround this. I will check. The 'cyclic' solver does not have this problem, but may be difficult to compute the null space in that setting.
> 
> An alternative is, if you are able to permute the columns in a way that A = [ B C ] with B square and non-singular, then the nullspace will be
> [ B^{-1}*C ]
> [   I_k    ]
> The difficult part here is to do the permutation, but you said your A matrix is already in this form.

This is another possibility. In fact, the matrix B you indicate is diagonalizable with the same unitary transformation for all the values of the external parameter Q. This is the method of choice when we can diagonalize the matrix. If one is not able to make the diagonalization, then I imagine this solution is still viable, but I don’t know how to get an inexpensive way of approximating the action of B^{-1} on C. Also, for some parameters, there may be a couple of eigenvalues of B that get very very close to zero.