Non-linear optimization: analysis of eigenvalues, subject to boundary conditions

For some non-linear finite element programs I have a matrix of tangent rigidity $$textbf {K} in mathbb {R} ^ {n times n}$$, which is symmetric. I want to find the Eigenvector corresponding to the smallest Eigenvalue of this matrix, so that each component of this Eigenvector is not negative. I know that the smallest Eigenpair can be found through different methods (for example, inverse iterative method, Rayleigh quotient iteration, Lanczos method, etc.). But is there a method to extract the proper pair of the matrix of tangent rigidity, so that all the components of the eigenvector are not negative? Is this possible?