About the matrix variate problem involved in the distribution of $\mathbf{E}^{-1}\mathbf{H}$

TL;DR

This paper investigates the distribution of the matrix product involving inverse and another matrix within the context of multivariate linear hypothesis testing, focusing on sums of squares and products related to hypotheses and errors.

Contribution

It provides a detailed analysis of the distribution of the nonsymmetric matrix product    involving inverse matrices in multivariate linear models, a topic with limited prior exploration.

Findings

Distribution formulas derived for    of  and  matrices.

Insights into the behavior of the matrix product under hypothesis testing scenarios.

Framework applicable to multivariate linear hypothesis problems.

Abstract

This work studies the distribution of the nonsymmetric matrix $\mathbf{E}^{-1}\mathbf{H}$. This random product is of fundamental interest under the general multivariate linear hypothesis setting. Specifically when $\mathbf{H}$ and $\mathbf{E}$ are seen as the sums of squares and the sums of products due to the hypothesis and due to the error, respectively.