Gaussian Distributions on the Space of Symmetric Positive Definite Matrices from Souriau’s Gibbs State for Siegel Domains by Coadjoint Orbit and Moment Map

Abstract

AbstractWe will introduce Gaussian distribution on the space of Symmetric Positive Definite (SPD) matrices, through Souriau’s covariant Gibbs density by considering this space as the pure imaginary axis of the homogeneous Siegel upper half space where Sp (2n,R)/U(n) acts transitively. Gauss density of SPD matrices is computed through Souriau’s moment map and coadjoint orbits. We will illustrate the model first for Poincaré unit disk, then Siegel unit disk and finally upper half space. For this example, we deduce Gauss density for SPD matrices.KeywordsSymmetric positive definite matricesLie groups thermodynamicsSymplectic geometryMaximum entropyExponential density family