The Commutation Matrix: Some Properties and Applications

1979/03/01 by Jan R. Magnus, H. Neudecker, Heinz Neudecker · 551 citations
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Bayesian Methods and Mixture Models #Combinatorics #Covariance matrix #Distribution (mathematics) #Geometry #Kronecker delta #Kronecker product #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Multivariate statistics #Order (exchange) #Physics #Product (mathematics) #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Statistics #Wishart distribution

paper · pdf · doi:10.1214/aos/1176344621

published in The Annals of Statistics 7(2) (Institute of Mathematical Statistics)

openalex publication_date 1979/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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 ε' Aε⋅ε' Bε⋅ε'Cε, where ε ∼ N(0, V), and A, B, C are symmetric. Further, the expectation and covariance matrix of x ⊗ y are found, where x and y are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

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