Abstract
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.
Highlights
The diagonalization of matrices may be the top priority in the application of modern physics
The question is, what would happen if the magnitude of matrix elements in some rows is significantly larger or smaller than those in other rows? Our results show that the linear relationship still keeps
Besides of direct approximations to ground state eigenvectors, the scaling relationship proposed in this article may be used to improve the performance of other numerical methods
Summary
The diagonalization of matrices may be the top priority in the application of modern physics. The relationship implies a straightforward method to directly calculate the eigenvector of ground states for one kind of matrices. One possible idea is to establish an immediate connection between eigenvectors and matrix elements If this kind of connection can be figured out, it may be an appealing method for matrix diagonalization. In recent studies on many-body quantum systems, the physical properties are predicted without explicitly diagonalizing Hamiltonian matrices[18,19,20,21,22,23,24,25]. Enlightened by many successful cases in the machine learning or Big Data analysis, we expect that, the connection between matrix elements and eigenvectors could be pried through the deep analysis for an enormous number of matrices. More multi-applications of RMs in physics can be found, for instance, in refs. 36–41
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