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Calibration Procedures for Orthogonal Superposition Rheology
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Published on: November 18, 2020

Truncations of random orthogonal matrices.

Boris A Khoruzhenko1, Hans-Jürgen Sommers, Karol Życzkowski

  • 1Queen Mary University of London, School of Mathematical Sciences, London E1 4NS, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary

This study investigates nonsymmetric real random matrices, revealing a unique eigenvalue distribution with both real and complex eigenvalues. Strong nonorthogonality leads to behavior seen in the real Ginibre ensemble.

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Area of Science:

  • Mathematics
  • Random Matrix Theory
  • Statistical Physics

Background:

  • Nonsymmetric real random matrices are crucial in various scientific fields, including quantum chaos and nuclear physics.
  • Understanding their statistical properties, particularly eigenvalue distributions, is key to modeling complex systems.

Purpose of the Study:

  • To investigate the statistical properties of nonsymmetric real random matrices derived from truncated orthogonal matrices.
  • To derive an exact formula for the density of eigenvalues for these matrices.
  • To analyze the behavior under different nonorthogonality conditions.

Main Methods:

  • Derivation of an exact formula for eigenvalue density.
  • Analysis of spectral properties under varying matrix dimensions (M and N).
  • Comparison with established random matrix ensembles like the real Ginibre ensemble.

Main Results:

  • An exact eigenvalue density formula was derived, featuring a mix of real eigenvalues and a spectrum within the unit disk, symmetric about the real axis.
  • In cases of strong nonorthogonality (M/N = constant), behavior consistent with the real Ginibre ensemble was observed.
  • For M = N-L with fixed L, a universal distribution of resonance widths was recovered.

Conclusions:

  • The study provides a comprehensive analytical framework for understanding the statistical properties of truncated random orthogonal matrices.
  • The findings reveal distinct spectral behaviors depending on the degree of nonorthogonality, bridging known ensembles.
  • This work contributes to the broader understanding of random matrix theory and its applications in physics and mathematics.