C++ Eigen LU decomposition - sign of determinant

Question

I use C++ 14 and Eigen, I want to compute the determinant of a square matrix using LU decomposition. (link) but I have some issues: A is the main matrix (n size) and rez is the LU form of A.

PartialPivLU<MatrixXd> rez = PartialPivLU<MatrixXd>(A);
MatrixXd r1 = rez.matrixLU().triangularView<UpLoType::Upper>();
double det_r1 = 1;
for(int i=0;i<n;i++)
    det_r1 = det_r1 * r1(i,i);
cout<<det_r1<<endl;
cout<<A.determinant();

Determinant of r1 is the product of the elements from main diagonal. The problem is that det(A) isn't equal to det(r1).

For exemple det(A) = 500 and det(r1) = -500. The problem is about sign of r1, how can I get the sign?

Answer 1

The very documentation you link to says this is LU factorization with partial pivoting. That means you're decomposing A as PLU, where P is a permutation matrix. Its determinant is the sign of the permutation.

From the documentation you link to:

This class represents a LU decomposition of a square invertible matrix, with partial pivoting: the matrix A is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P is a permutation matrix.

In your case, the permutation matrix must represent an odd permutation, and therefore has determinant -1.