The Commutation Matrix: Some Properties and Applications

1979/03/01 by Jan R. Magnus, H. Neudecker, Heinz Neudecker · 19 citations
Mathematics · Computer Science · #Advanced Statistical Methods and Models #Random Matrices and Applications #Bayesian Methods and Mixture Models

paper · pdf · 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 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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• Commutation matrices and Commutation tensors
• Commutators in finite free probability, I
• Covariance as a commutator
• Random commuting matrices
• Permutation-Like Matrices