I am trying to apply Cholesky Decomposition on a symmetric, voluntarily non positive definite matrix, to see how the system behaves. The matrix I am analysing is a 3*3 matrix whose eigenvalues are: -29.5, 2, 30.5. Since there is a negative eigenvalue, the matrix is not positive definite. Despite this, by applying the llt() method offered by the Eigen library, I still manage to obtain a Cholesky decomposition of my matrix. What I would like to know is if Eigen accidentally uses alternative methods (for example pseudo inverse or pseudo determinant) to solve the problem anyway. Can we say that the llt() method works (albeit with stability problems) even for non positive definite matrices?