The 70th anniversary of the distribution of random matrices: A survey
The author provides a survey of the last \(70\) years of the distribution of a variety of matrices that arise from matrix factorizations. He begins in the year 1928 with the distribution of \textit{J. Wishart} [Biometrika 20 (A), 32-52 (1928; JFM 54.0565.02)], that arises from the distribution of the covariance matrix of a sample from a standard multivariate normal distribution. Some cases studied are the factorization in rectangular coordinates, \( X=TG,S=TT^{\prime }\), where \(X\) is a \(p\times n\) random matrix, \(S\) is a positive definite symmetric matrix, \(T\) is a \(p\times p\) lower triangular matrix, \(G\) is a \(p\times n\) suborthogonal matrix, that is, \(GG^{\prime }=I_{p}\), the eigenvalue decomposition, the singular value decomposition \( X=GD_{s}H\), where \(G\) is orthogonal, \(H\) is a suborthogonal \(p\times n\) and \( D_{s}= \text{diag}(s_{1},\ldots ,s_{p})\), where \(s_{i}\) are the singular values, the \(LU\) decomposition, \(X=LU\) (asymmetric decomposition) and \(X=LD_{\theta }U\) (symmetric decomposition) and the triangular factorization \(X=GTG^{\prime }\) where \(G\) is an orthogonal matrix and \(T\) is a lower triangular matrix. He\ also describes the density of the Moore-Penrose inverse of \(X\), \( X^{+}=(XX^{\prime })^{-1}X\), the distribution of the inverse of \(X\), and the distribution of the square and square root of a symmetric positive definite matrix.
scientific article; zbMATH DE number 1864726
scientific article