  In this paper, we propose an unconstrained framework for eigenvalue problems
in both discrete and continuous settings. We begin our discussion to solve a
generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ with two $N\times
N$ real symmetric matrices $A, B$ via minimizing a proposed functional whose
nonzero critical points ${\bf x}\in\mathbb{R}^N$ solve the eigenvalue problem
and whose local minimizers are indeed global minimizers. Inspired by the
properties of the proposed functional to be minimized, we provide analysis on
convergence of various algorithms either to find critical points or local
minimizers. Using the same framework, we will also present an eigenvalue
problem for differential operators in the continuous setting. It will be
interesting to see that this unconstrained framework is designed to find the
smallest eigenvalue through matrix addition and multiplication and that a
solution ${\bf x}\in\mathbb{R}^N$ and the matrix $B$ can compute the
corresponding eigenvalue $\lambda$ without using $A$ in the case of $A{\bf
x}=\lambda B{\bf x}$. At the end, we will present a few numerical experiments
which will confirm our analysis.
