The Annals of Statistics

The Commutation Matrix: Some Properties and Applications

Jan R. Magnus and H. Neudecker

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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.

Article information

Ann. Statist., Volume 7, Number 2 (1979), 381-394.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 15A69: Multilinear algebra, tensor products
Secondary: 62H99: None of the above, but in this section

Stochastic vectors Kronecker product expectations covariance matrices


Magnus, Jan R.; Neudecker, H. The Commutation Matrix: Some Properties and Applications. Ann. Statist. 7 (1979), no. 2, 381--394. doi:10.1214/aos/1176344621.

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