An overview of recent developments in functions of matrix argument (Q6619563)

The main purpose of this paper is to give an overview of current research in some topics concerning the functions of a matrix argument. The author considers a real-valued scalar function \(f(X)\), where the argument \(X\) may be scalar, vector, matrix or a sequence of matrices in the real or complex domain. The function \(f(X)\), associated with \(X\), may be a function of \(XX^T\). By imposing additional conditions, \(f(X)\) can also be a statistical density of \(X\). All the models introduced in this paper are nonsingular in the sense that when square matrices are involved, then they are nonsingular and when a rectangular matrix \(X\) is involved, then \(XX^T\) is nonsingular. The text is accompanied by several illustrative examples.\N\NIndividual sections deal with the following topics:\N\begin{itemize}\N\item Evaluation of some general integrals and introduction of some general densities, mainly of the matrix-variate type. \N\N\item Details about some recent procedures involving logistic type models for vector/matrix variables in the real and complex domains.\N\N\item Evaluation of the normalizing constants in some general matrix-variate densities and introduction of some general scaling models and general Bayesian structures.\N\N\item Connection of Mellin convolutions of products and ratios and statistical distribution theory to fractional calculus, especially to fractional integrals of the first and second kinds.\N\N\item Derivations of matrix-variate functions through optimization of Mathai entropy.\N\N\item A glimpse into singular matrix-variate gamma and beta functions.\N\end{itemize}