Establishing simple relationship between eigenvector and matrix elements

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 a(−S_i)^p+c, where S_i is the sum of elements in the i-th row of the matrix with p, a 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 provides the optimal initial state for other more accurate methods, such as the Lanczos method.