The Commutation Matrix: Some Properties and Applications

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 ' A' B'C, where 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.