On matrices of random variables. General properties (Q1206001)

A matrix of random variables is studied as a special multivariate distribution by its characteristic matricial structure. Arranging the elements of a random variable matrix in a single vector, two covariance matrices can be defined, according to the arrangement of the elements by rows or by columns; these matrices are similar. Then, linear transformations are considered, in particular showing that a relation of affinity or equivalence between two random variables matrices involves a relation of congruence between the related covariance matrices. It is proved that, among the random variables matrices that are affine to a given one, there are distinct infinities with uncorrelated elements, with only uncorrelated columns or with only uncorrelated rows. Examples are given. A matrix sum of the covariance matrices of the column vectors of random variables is introduced and likewise for the row vectors, on the basis of which it is proved that there are distinct infinities, among equivalent random variables matrices, having columns or rows with uncorrelated variables. Further, it is proved that within the orthogonal transformations there is one single matrix with uncorrelated variables, one single matrix having columns with uncorrelated variables and a single one having rows with uncorrelated variables. Choosing the units of measure for these matrices suitably, distinct kinds of standardization are obtained. Finally, as a multivariate distribution may be represented as a linear function of its principal components or only of a part of them (common or latent factors), so a random variables matrix may be represented by similar ways.