The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form: I am considering linear elastic material constitutive. I checked the condition number which is order 10e3 indicating the augmented matrix is not much ill-conditioned. However, there are some negative eigenvalues for the matrix. What are possible reasons for the lack of semi-definite positiveness? It should be always positive semi-definite for the (linearized) global stiffness matrix whatever Lagrange multipliers method is employed. Is my Lagrange multipliers discretization/approximation space is not well set up or I lost something very important? Anyone can share some experience?