Establishing simple relationship between eigenvector and matrix elements

Abstract

A simple approximate relationship between the ground-state eigenvector and the sum of matrix elements in each row has been established for real symmetric matrices with non-positive off-diagonal elements. Specifically, the $i$-th components of the ground-state eigenvector could be calculated by $(-S_i)^p+c$, where $S_i$ is the sum of elements in the $i$-th row of the matrix with $p$ and $c$ being variational parameters. The simple relationship provides a straightforward method to directly calculate the ground-state eigenvector for a matrix. Our preliminary applications to the Hubbard model and the Ising model in a transverse field show encouraging results.The simple relationship also provide the optimal initial state for other more accurate methods, such as the Lanczos method.