Random matrix theory and symmetric spaces

Random matrix theory and symmetric spaces

In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important resu...

Lie Groups. Representation Theory and Symmetric Spaces

Determinant preserving transformations on symmetric matrix spaces

Let Sn(F) be the vector space of n × n symmetric matrices over a field F (with certain restrictions on cardinality and characteristic). The transformations φ on the space which satisfy one of the following conditions: 1. det(A+ λB) = det(φ(A) + λφ(B)) for all A,B ∈ Sn(F) and λ ∈ F; 2. φ is surjective and det(A+ λB) = det(φ(A) + λφ(B)) for all A,B and two specific λ; 3. φ is additive and preserv...

Random Matrix Theory and Applications

This summary will briefly describe some recent results in random matrix theory and their applications. 1 Motivation 1.1 Multiple Antenna Gaussian Channels 1.1.1 The deterministic case Consider a gaussian channel with t transmitting and r receiving antennas. The received vector y ∈Cr will depend on the transmitted vector x ∈Ct by y = Hx+n where H is an rxt complex matrix of gains and n is a vect...

Random matrix theory and number theory

We review some of the connections between Random Matrix Theory and Number Theory. These include modelling the value distributions of the Riemann zeta-function and other L-functions, and the statistical distribution of their zeros. 1.