An unconstrained framework for eigenvalue problems

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.