Weighted distributions of eigenvalues

Abstract

In this article, the weighted version of a probability density function is considered as a mapping of the original distribution. Generally, the properties of the distribution of a random matrix and the distributions of its eigenvalues are closely related. Therefore, the weighted versions of the distributions of the eigenvalues of the Wishart distribution are introduced and their properties are discussed. We propose the concept of rotation invariance for the weighted distributions of the eigenvalues of the Wishart and non-central Wishart distributions. We also introduce here, the concept of a “mirror”, meaning, looking at the distribution of a random matrix through the distribution of its eigenvalues. Some graphical representations are given, to visualize the weighted distributions of the eigenvalues for specific cases.