This paper deals with matrix-variate distributions, from Wishart to Inverse Elliptical Wishart distributions over the set of symmetric definite positive matrices. Similar to the multivariate scenario, (Inverse) Elliptical Wishart distributions form a vast and general family of distributions, encompassing, for instance, Wishart or $t$-Wishart ones. The first objective of this study is to present a unified overview of Wishart, Inverse Wishart, Elliptical Wishart, and Inverse Elliptical Wishart distributions through their fundamental properties. This involves leveraging the stochastic representation of these distributions to establish key statistical properties of the Normalized Wishart distribution. Subsequently, this enables the computation of expectations, variances, and Kronecker moments for Elliptical Wishart and Inverse Elliptical Wishart distributions. As an illustrative application, the practical utility of these generalized Elliptical Wishart distributions is demonstrated using a real electroencephalographic dataset. This showcases their effectiveness in accurately modeling heterogeneous data.

翻译：本文研究矩阵变量分布，范围从Wishart分布到对称正定矩阵集上的逆椭圆型Wishart分布。与多变量场景类似，（逆）椭圆型Wishart分布构成一个庞大且通用的分布族，例如涵盖Wishart分布或t-Wishart分布。本研究的第一目标是基于这些分布的基本性质，对Wishart分布、逆Wishart分布、椭圆型Wishart分布和逆椭圆型Wishart分布进行统一概述。这涉及利用这些分布的随机表示来建立归一化Wishart分布的关键统计性质。随后，此方法能够计算椭圆型Wishart分布和逆椭圆型Wishart分布的期望、方差及Kronecker矩。作为应用实例，通过真实脑电图数据集展示了这些广义椭圆型Wishart分布的实际效用，验证了其在精确建模异质数据方面的有效性。