Pan Wei, Wang Jing, Sun Deyan
Department of Physics, East China Normal University, 200241, Shanghai, China.
Sci Rep. 2020 Feb 25;10(1):3414. doi: 10.1038/s41598-020-60103-5.
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 relationship 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 random matrices, the linear relationship still keeps for non-random matrices, in which off-diagonal elements are non-positive. The relationship implies a straightforward method to directly calculate the eigenvector of ground states for one kind of matrices. The tests on both Hubbard and Ising models show that, this new method works excellently.
矩阵对角化可能是现代物理学应用中的首要任务。在本文中，我们通过数值方法证明，对于非对角元素为非正值的实对称随机矩阵，本征向量与矩阵元素之间存在一种通用的标度关系。也就是说，基态本征向量的每个元素与相应行中矩阵元素的总和呈线性相关。尽管该结论是基于随机矩阵得出的，但对于非对角元素为非正值的非随机矩阵，这种线性关系仍然成立。这种关系意味着一种直接计算一类矩阵基态本征向量的简便方法。对哈伯德模型和伊辛模型的测试表明，这种新方法效果极佳。
