A pathway to matrix-variate gamma and normal densities

A general real matrix-variate probability model is introduced here, which covers almost all real matrix-variate densities used in multivariate statistical analysis. Through the new density introduced here, a pathway is created to go from matrix-variate type-1 beta to matrix-variate type-2 beta to matrix-variate gamma to matrix-variate Gaussian or normal densities. Other densities such as extended matrix-variate Student t, F, Cauchy density will also come in as particular cases. Connections to the distributions of quadratic forms and generalized quadratic forms in the new matrix are established. The present day analysis of these problems is mainly confined to Gaussian random variables. Thus, through the new distribution, all these theories are extended. Connections to certain geometrical probability problems, such as the distribution of the volume of a random parallelotope in Euclidean space, is also established.