Eigen: Obtain the kernel of a sparse matrix - c++

Question

Given a sparse matrix A and a vector b, I would like to obtain a solution x to the equation A * x = b as well as the kernel of A.
One possibility is to convert A to a dense representation.
#include <iostream>
#include <Eigen/Dense>
#include <Eigen/SparseQR>
int main()
{
// This is a toy problem. My actual matrix
// is of course bigger and sparser.
Eigen::SparseMatrix<double> A(2,2);
A.insert(0,0) = 1;
A.insert(0,1) = 2;
A.insert(1,0) = 4;
A.insert(1,1) = 8;
A.makeCompressed();
Eigen::Vector2d b;
b << 3, 12;
Eigen::SparseQR<Eigen::SparseMatrix<double>,
Eigen::COLAMDOrdering<int> > solver;
solver.compute(A);
std::cout << "Solution:\n" << solver.solve(b) << std::endl;
Eigen::Matrix2d A_dense(A);
std::cout << "Kernel:\n" << A_dense.fullPivLu().kernel() << std::endl;
return 0;
}
Is it possible to do the same directly in the sparse representation? I could not find a function kernel() anywhere except in FullPivLu.

Answer 1

I think #chtz's answer is almost correct, except we need to take the last A.cols() - qr.rank() columns. Here is a mathematical derivation.
Say we do a QR decomposition of your matrix Aᵀ as
Aᵀ * P = [Q₁ Q₂] * [R; 0] = Q₁ * R
where P is the permutation matrix, thus
Aᵀ = Q₁ * R * P⁻¹.
We can see that Range(Aᵀ) = Range(Q₁ * R * P⁻¹) = Range(Q₁) (because both P and R are invertible).
Since Aᵀ and Q₁ have the same range space, this implies that A and Q₁ᵀ will also have the same null space, namely Null(A) = Null(Q₁ᵀ). (Here we use the property that Range(M) and Null(Mᵀ) are complements to each other for any matrix M, hence Null(A) = complement(Range(Aᵀ)) = complement(Range(Q₁)) = Null(Q₁ᵀ)).
On the other hand, since the matrix [Q₁ Q₂] is orthonormal, Null(Q₁ᵀ) = Range(Q₂), thus Null(A) = Range(Q₂), i.e., kernal(A) = Q₂.
Since Q₂ is the right A.cols() - qr.rank() columns, you could call rightCols(A.cols() - qr.rank()) to retrieve the kernal of A.
For more information on kernal space, you could refer to https://en.wikipedia.org/wiki/Kernel_(linear_algebra)