Spherical ensembles
The authors give a unifying study regarding densities of random matrices with entries being real (\(\beta=1\)), complex (\(\beta=2\)), quaternion (\(\beta=4\)) or octonion (\(\beta=8\)) numbers. The kind of matrices studied in the paper are characterized by the so-called spherical and elliptical distributions, the first ones are defined through an invariant property when multiplying, for example in the complex case, by unitary matrices; and the second ones are defined through a special form of the distribution on the space of matrices. One instance of this kind of ensembles is when imposing independence of the entries, in which case the entries should be normal random variables. The first results describe the distributions of the ensembles and the so-called joint element densities. The authors then turn into the study of certain constructions of the form \((M+M^{*})/2\) or \(M^{*}M\), where \(M^{*}\) represets the conjugate transpose matrix; this is related to what is known as the Hermite, Laguerre, Jacobi and Fourier ensembles, and the authors use the term vector-spherical ensembles in their definition. In turn, they are able to describe eigenvalue densities of these random matrices. In the last part of the paper, the authors also describe densities associated to some symmetric tridiagonal random matrices related to their vector-spherical ensembles, with the parameter \(\beta\) belonging to \([0,\infty)\).
scientific article; zbMATH DE number 6153403
scientific article