Eigen - Check if matrix is Positive (Semi-)Definite

Question

I'm implementing a spectral clustering algorithm and I have to ensure that a matrix (laplacian) is positive semi-definite.

A check if the matrix is positive definite (PD) is enough, since the "semi-" part can be seen in the eigenvalues. The matrix is pretty big (nxn where n is in the order of some thousands) so eigenanalysis is expensive.

Is there any check in Eigen that gives a bool result in runtime?

Matlab can give a result with the chol() method by throwing an exception if a matrix is not PD. Following this idea, Eigen returns a result without complaining for LLL.llt().matrixL(), although I was expecting some warning/error. Eigen also has the method isPositive, but due to a bug it is unusable for systems with an old Eigen version.

Answer 1

You can use a Cholesky decomposition (LLT), which returns Eigen::NumericalIssue if the matrix is negative, see the documentation.

Example below:

#include <Eigen/Dense>

#include <iostream>
#include <stdexcept>

int main()
{
    Eigen::MatrixXd A(2, 2);
    A << 1, 0 , 0, -1; // non semi-positive definitie matrix
    std::cout << "The matrix A is" << std::endl << A << std::endl;
    Eigen::LLT<Eigen::MatrixXd> lltOfA(A); // compute the Cholesky decomposition of A
    if(lltOfA.info() == Eigen::NumericalIssue)
    {
        throw std::runtime_error("Possibly non semi-positive definitie matrix!");
    }    
}