## The Annals of Statistics

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

### The Commutation Matrix: Some Properties and Applications

Jan R. Magnus and H. Neudecker

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aos/1176344621

**Mathematical Reviews number (MathSciNet)**

MR520247

**Zentralblatt MATH identifier**

0414.62040

**JSTOR**

links.jstor.org

**Subjects**

Primary: 15A69: Multilinear algebra, tensor products

Secondary: 62H99: None of the above, but in this section

**Keywords**

Stochastic vectors Kronecker product expectations covariance matrices

#### Citation

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