Random Matrix

A Random Matrix is a random element that is a matrix composed of random variables.

• • …
• Counter-Example(s):
  • Random Tree.
  • Random Vector.
• See: Gaussian Unitary Ensemble, Wishart Random Matrix, Hermitian Matrix, System of Linear Equations.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/random_matrix Retrieved:2015-2-16.
  • In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

• http://en.wikipedia.org/wiki/Random_element#Random_matrix
  • A random matrix is a matrix-valued random element. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

2012

• (Tao, 2012) ⇒ Terence Tao. (2012). “Topics in Random Matrix Theory." American Mathematical Soc., Vol. 13.

2011

• (2011). “ The Oxford Handbook of Random Matrix Theory." Oxford University Press.
  • BOOK OVERVIEW: In part one, all modern and classical techniques of solving random matrix models are explored, including orthogonal polynomials, exact replicas or supersymmetry. Further, all main extensions of the classical Gaussian ensembles of Wigner and Dyson are introduced including sparse, heavy tailed, non-Hermitian or multi-matrix models. In the second and larger part, all major applications are covered, in disciplines ranging from physics and mathematics to biology and engineering. This includes standard fields such as number theory, quantum chaos or quantum chromodynamics, as well as recent developments such as partitions, growth models, knot theory, wireless communication or bio-polymer folding.
  • QUOTE: Before we describe some general features of random matrix theory (RMT) we would like to first introduce the simplest and maybe most frequently used standard example, the Gaussian unitary ensembles (GUE) of random matrices.[1]

2009

• (Zeng & Liang, 2009) ⇒ Yonghong Zeng, and Ying-Chang Liang. (2009). “Eigenvalue-based Spectrum Sensing Algorithms for Cognitive Radio.” In: IEEE Transactions on Communications, 57(6).
  • QUOTE: … Rη(Ns) = 1 Ns L−2+Ns ∑ n=L−1 ˆη(n)ˆη† (n). (23) Rη(Ns) is nearly a Wishart random matrix [18]. The study of the spectral (eigenvalue distributions) of a random matrix is a very hot topic in recent years in mathematics as well as communication and physics. …

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