# A density-free approach to the matrix variate beta distribution

Mitra, Sujit Kumar (1970) A density-free approach to the matrix variate beta distribution Sankhya - Series A, 32 (1). pp. 81-88. ISSN 0581-572X

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## Abstract

Let \$ S_1\sim W_k(n_1;\boldsymbol\Sigma)\$ and \$ S_2\sim W_k(n_2,\boldsymbol\Sigma)\$ be independent Wishart matrices and \$\boldsymbol\Sigma\$ be p.d. Consider \$ S= S_1+ S_2\$ and define \$ U=(S)ˆ{-½} S_1[( S)ˆ{-½}]\$, where \$( S)ˆ{½}\$ is the unique lower triangular matrix with positive diagonal elements, such that \$( S)ˆ{½}[( S)ˆ{½ }]'= S\$, and \$( S)ˆ{-½}\$ is the inverse of \$( S)ˆ{½}\$. The joint distribution of the element of the symmetric matrix \$ U\$ is said to be `matrix variate beta' and is denoted by the symbol \$B_k(n_½,n_2/2)\$. Several interesting properties of this distribution are obtained in this paper.

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