On a matrix identity associated with generalized least squares
type: Item
instance of: scholarly article
title: On a matrix identity associated with generalized least squares
zbMATH Open document ID: 0694.15004
DOI: 10.1016/0024-3795(90)90355-G
author: F. R. De Hoog
author: Terence P. Speed
author: Emlyn R. Williams
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: For reduced generalized least squares equations with a heteroscedastic dispersion matrix V, the following two matrix identities are proved: \[ V^{-1}-V^{-1}X(X^ TV^{-1}X)^-X^ TV^{-1}=\{[I-X(X^ TX)^- X^ T]V[I-X(X^ TX)^--X^ T]\} \] and \[ V^{-1}(I-Q)=[(I-P)V(I- P)]^+ \] where \(A^-\) and \(A^+\) denote an arbitrary and the Moore- Penrose generalized inverse respectively and \(A^ T\) denotes the transpose of the matrix A. Here P is the projection \(P=X(X^ TX)^-X^ T\) where X is an (n\(\times p)\) matrix and V a positive definite dispersion matrix. Illustration of these identities are developed for the regression model \(Ey=X\beta +Z\gamma,\) \(Dy=V\) where E and D are the expectation and dispersion operators for the random (n\(\times 1)\) vector y.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 4136123
zbMATH Keywords: reduced generalized least squares
zbMATH Keywords: heteroscedastic dispersion matrix
zbMATH Keywords: matrix identities
zbMATH Keywords: Moore-Penrose generalized inverse
zbMATH Keywords: regression
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4136123