Hi,
i'm using Eigen3 to compute the inverse matrix of a covariances matrix P.
Tipically P is singular so I compute the inverse (or pseudo-inverse) using SVD decomposition and the code used in this post : http://eigen.tuxfamily.org/bz/show_bug.cgi?id=257.
Apparently it works well but i have a problem.
Just for check, before computing the pseudo-inverse, I print out the determinant and the rank of the matrix P using this code :

Eigen::FullPivLU<Eigen::MatrixXd> lu_decomp(P);
std::cout << Determinant is " << A.determinant() << std::endl;
std::cout << "The rank of P is " << lu_decomp.rank() << std::endl;
std::cout << "The size of P is " << P.rows() " x " << P.cols() << std::endl;

In some cases (most of them) I obtain for example:

Detrminant is 0
The rank of P is 60
The size of P is 60x60

or

Detrminant is -0
The rank of P is 72
The size of P is 72x72

I think it is wrong, because if I have a full rank matrix its determinant should not be zero. Any suggest ?

Thank's All.
Angelo.

This doesn't directly answer your question but a covariance matrix is positive semidefinite so wouldn't you be better off using a LDLT decomposition instead of an LU? The LDLT decomposition is not necessarily rank-revealing but the number of diagonal elements greater than a threshold times the largest diagaonal element is a good indication of the rank. The determinant of A is simply the product of the diagonal elements of D, which are retrieved with the .vectorD() method.

Perhaps the computation of the determinant underflow. For instance imagine a 50x50 diagonal matrix with all the elements around 1e-8. Without additional information, i.e., a relevant non zero reference value, the matrix is full rank but the determinant is 1e-(8*50) which is exactly 0 using double precision numbers.

Compute the log-determinant instead. Perform Cholesky factorisation and sum the elements: $\log \det (A) = \sum_{i=0}^{n-1} 2*\log(L_{ii})$. After that exponentiate the result to get $\det(A)$