# A new strategy for directly calculating the minimum eigenvector of matrices without diagonalization

## TL;DR

This paper introduces a novel, direct method for calculating the ground state eigenvector of certain matrices by exploiting a linear relationship with matrix row sums, bypassing traditional diagonalization techniques.

## Contribution

The paper presents a universal linear relationship between eigenvector elements and row sums for specific matrices, enabling a new direct calculation method for ground state eigenvectors.

## Key findings

- The linear relationship holds for real symmetric matrices with non-positive off-diagonal elements.
- The method accurately computes ground state eigenvectors in Hubbard and Ising models.
- The approach bypasses traditional diagonalization, simplifying eigenvector calculations.

## 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 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.