  This paper is concerned with the P1 finite element approximation of the
eigenvalue problem of second-order elliptic operators subject to the Dirichlet
boundary condition. The focus is on the preservation of basic properties of the
principal eigenvalue and eigenfunctions of continuous problems. It is shown
that when the stiffness matrix is an irreducible $M$-matrix, the algebraic
eigenvalue problem maintains those properties such as the smallest eigenvalue
being real and simple and the corresponding eigenfunctions being either
positive or negative inside the physical domain. Mesh conditions leading to
such a stiffness matrix are also studied. A sufficient condition is that the
mesh is simplicial, acute when measured in the metric specified by the inverse
of the diffusion matrix, and interiorly connected. The acute requirement can be
replaced by the Delaunay condition in two dimensions. Numerical results are
presented to verify the theoretical findings.
