Commutation matrix

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.

Abstract: The vec-permutation matrix I m,n is defined by the equation vec A m × n = I m,n vecA′, Where vec is the vec operator such that vecA is the vector of columns of A stacked one under the other. The variety of definitions, names and notations for I m,n are discussed, and its properties are developed by simple proofs in contrast to certain lengthy proofs in the literature that are based on descriptive definitions. For example, the role of I m,n in reversing the order of Kronecker products is succinctly derived using the vec operator. The matrix M m,n is introduced as M m,n = I m,n M; it is the matrix having as rows,every nth row starting with the first, then every nth row starting with the second, and so on. Special cases of M m,n are discussed.

Abstract: A new way of solving the matrix equation AX + XB = C is presented in the form (Vec X) = f(A, B).(Vec C). The solution makes use of the properties of the Kronecker products and is particularly suitable for finding X in terms of a fixed set of A and B and a varying C.

Abstract: where B' is the transpose of B. It has been shown that Definition 1 and Theorem 1 can fruitfully be applied to problems of matrix differentiation [2]. In this note it will be shown that they can be applied to a more general class of linear matrix equations, including linear matrix differential equations. Firstly, four standard properties of Kronecker products have to be related, all of which may be proved in an elementary fashion [1, p. 223 if.]. The matrices involved can have any appropriate orders. In Property 4 it is assumed that A and B are square of order m and s, respectively. (The same order assumption will be made in Theorems 2 and 3, which will be presented further on.) PROPERTY 1. (A B)(C D) =(AC) (BD). PROPERTY 2. (A 0 B)' = A' 0 B'. PROPERTY. 3. (A + B) 09 (C + D) = A C + A D + B (& C + B X D. PROPERTY 4. If A has characteristic roots oci, i = 1, -*, m, and if B has characteristic roots /IB, j = 1,... , s, then A 0 B has characteristic roots oci,Bj. Further, Is 0 A + B 0) Im has characteristic roots oci + f3B. For the treatment of differential equations we need the matrix exponential