## The Annals of Statistics

### The Commutation Matrix: Some Properties and Applications

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176344621

Digital Object Identifier
doi:10.1214/aos/1176344621

Mathematical Reviews number (MathSciNet)
MR520247

Zentralblatt MATH identifier
0414.62040

JSTOR