On the universality of the probability distribution of the product B -1X of random matrices

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Abstract

Consider random matrices A, of dimension m × (m + n), drawn from an ensemble with probability density f(tr AA†), with f(x) a given appropriate function. Break A = (B, X) into an m × m block B and the complementary m × n block X, and define the random matrix Z = B -1X. We calculate the probability density function P(Z) of the random matrix Z and find that it is a universal function, independent of f(x). The universal probability distribution P(Z) is a spherically symmetric matrix-variate t-distribution. Universality of P(Z) is, essentially, a consequence of rotational invariance of the probability ensembles we study. As an application, we study the distribution of solutions of systems of linear equations with random coefficients, and extend a classic result due to Girko.

Original languageEnglish
Pages (from-to)6823-6835
Number of pages13
JournalJournal of Physics A: Mathematical and General
Volume37
Issue number26
DOIs
StatePublished - 2 Jul 2004

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy (all)

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