I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully negative definite for a local minimum)
But since I have a system of 1000 variables, I am stuck with a 1000*1000 matrix to determine whether it is positive definite or negative definite.(this is all real numbers)
Is there any way to do it other than calculates all the eigenvalue ? Someone suggests a cholesky decomposition, but I failed to understand why.