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March, 1979 The Commutation Matrix: Some Properties and Applications
Jan R. Magnus, H. Neudecker
Ann. Statist. 7(2): 381-394 (March, 1979). DOI: 10.1214/aos/1176344621

Abstract

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

Citation

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Jan R. Magnus. H. Neudecker. "The Commutation Matrix: Some Properties and Applications." Ann. Statist. 7 (2) 381 - 394, March, 1979. https://doi.org/10.1214/aos/1176344621

Information

Published: March, 1979
First available in Project Euclid: 12 April 2007

zbMATH: 0414.62040
MathSciNet: MR520247
Digital Object Identifier: 10.1214/aos/1176344621

Subjects:
Primary: 15A69
Secondary: 62H99

Keywords: Covariance matrices , expectations , Kronecker product , Stochastic vectors

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 2 • March, 1979
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