On second-order and fourth-order moments of jointly distributed random matrices: A survey
type: Item
instance of: scholarly article
title: On second-order and fourth-order moments of jointly distributed random matrices: A survey
zbMATH Open document ID: 1033.15016
DOI: 10.1016/S0024-3795(00)00181-6
author: Ghazal A. Ghazal
author: Heinz Neudecker
published in: Linear Algebra and its Applications
publication date: 2004-03-29T00:00:00Z
review text: Random matrices \(Z=(X,Y)\) are considered, where the matrices \(X\) and \(Y\) have identical row numbers and normal jointly distributed entries. The paper presents second-order and fourth-order moments of \(Z\) in the tensor form in terms of expectations \(EX,\;EY\) and generalized covariance matrices \(\Omega=\{\Omega_{XX}, \Omega_{XY},\Omega_{YX},\Omega_{YY}\}\), where \(\Omega_{AB}\overset{def}{=}\;\text{cov} (\text{vec} A,\text{ vec} B),\) and each matrix \(A\) is considered as a generalized vector \(\text{vec} A =(A_{.1},\dots,A_{.n})\), where \(A_{.j}\) is the corresponding subset of entries \(\{A_{ij}\}\). The tensor relations are obtained for \(E(X t X),\;E(X' t X),\;E(X t Y)\), where the prime denotes transposing, and also for covariance matrices of \({\text{ vec}} (X t Y),\) \({\text{vec}}(X' t Y)\), \( {\text{ vec}}(X t X),\) \({\text{ vec}} (X' t X).\) Two lemmas are proved stating the identity of some of these expressions. A number of special cases is considered. The tensor expressions \(E(X'AY),\;E(XAY'),\;E(XAY),\;E(X'AY')\) are evaluated and tensor expressions are obtained for expectations and covariances of quadratic forms \(S_A=X'AX\) and \(S_B=X'BX\). As a special case, the expressions \(E(SBS)\) are evaluated for the matrices \(S\) having the Wishart distribution.
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 1563480
zbMATH Keywords: random matrices
zbMATH Keywords: moments
zbMATH Keywords: covariance matrices
zbMATH Keywords: quadratic forms
zbMATH Keywords: Wishart distribution
MaRDI profile type: Publication
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Wikidata QID: Q127884555
description: scientific article; zbMATH DE number 1563480