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.

The commutation matrix: Some properties and applications

Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Solving a nonlinear fractional differential equation using Chebyshev wavelets

The Legendre wavelet method for solving fractional differential equations

Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet

Kronecker operational matrices for fractional calculus and some applications

Vector least-squares solutions for coupled singular matrix equations

This paper is concerned with a long list of matrix products for partitioned and non-partitioned matrices, including Kronecker, Khatri-Rao of first kind, Hadamard, Tracy-Singh, Khatri-Rao, block Kronecker, block Hadamard,and box products (sums). Our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place. Several new attractive connections are also added to this collection. We give, as an example, two applications in order to show that these connections play a central role in many applications. These applications are: A block diagonal least-squares problem and a generalization of matrix inequalities involving Khatri-Rao products of positive definite matrices.

Some new connections between matrix products for partitioned and non-partitioned matrices

Recently, there have been many authors, who established a number of inequalities involving Khatri-Rao and Hadamard products of two positive matrices. In this paper, the results are established in the following three ways. First, we find generalization of the inequalities involving Khatri-Rao product using results given by Liu (1999), Mond and Pecaric (1997), Cao et al. (2002), Chollet (1997), and Visick (2000). Second, we recover and develop some results of Visick. Third, the results are extended to the case of Khatri-Rao product of any finite number of matrices. These results lead to inequalities involving Hadamard product, as a special case.

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Extension and generalization inequalities involving the Khatri-Rao product of several positive matrices

The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadamard and generalized Kronecker products, respectively. We define the KhatriRao and Tracy-Singh sums for partitioned matrices as generalized Hadamard and generalized Kronecker sums and derive some results including matrix equalities and inequalities involving the two sums. Based on the connection between the Khatri-Rao and Tracy-Singh products (sums) and use mainly Liu’s, Mond and Pečarić’s methods to establish new inequalities involving the Khatri-Rao product (sum). The results lead to inequalities involving Hadamard and Kronecker products (sums), as a special case.

/pdf/matrix-equalities-and-inequalities-involving-khatri-rao-and-21f87mz8qv.pdf

Zeyad Al Zhour

Papers

Kronecker operational matrices for fractional calculus and some applications

Vector least-squares solutions for coupled singular matrix equations

Some new connections between matrix products for partitioned and non-partitioned matrices

Extension and generalization inequalities involving the Khatri-Rao product of several positive matrices

Matrix equalities and inequalities involving khatri-rao and tracy-singh sums