  The diagonalization of matrices may be the top priority in the application of
modern physics. In this paper, we numerically demonstrate that, for real
symmetric random matrices with non-positive off-diagonal elements, a universal
scaling between the eigenvector and matrix elements exists. Namely, each
element of the eigenvector of ground states linearly correlates with the sum of
matrix elements in the corresponding row. Although the conclusion is obtained
based on the random matrices, the linear relationship still keeps for regular
matrices, in which off-diagonal elements are non-positive. The relationship
implies a straightforward method to directly calculate the eigenvector of
ground states for a kind of matrices. The test on both Hubbard and Ising models
shows that, this new method works excellently.
