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Area of Science:

  • Quantum Physics
  • Computational Physics
  • Linear Algebra

Background:

  • Matrix diagonalization is crucial in modern physics.
  • Understanding eigenvector properties is key for solving complex physical systems.

Purpose of the Study:

  • To numerically demonstrate a universal scaling relationship for eigenvectors.
  • To establish a linear correlation between eigenvector elements and matrix row sums.
  • To develop a novel method for calculating ground state eigenvectors.

Main Methods:

  • Numerical analysis of real symmetric random matrices.
  • Investigation of matrices with non-positive off-diagonal elements.
  • Testing the derived relationship on Hubbard and Ising models.

Main Results:

  • A universal linear scaling relationship between eigenvector elements and row sums was identified.
  • This relationship holds for both random and non-random matrices with non-positive off-diagonal elements.
  • The method accurately calculates ground state eigenvectors for tested models.

Conclusions:

  • The discovered linear relationship provides a direct method for eigenvector calculation.
  • This finding simplifies matrix diagonalization in specific physics applications.
  • The method's efficacy is validated on Hubbard and Ising models.