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 a(−Si)p+c, where Si 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.
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