Generalizations and Some Applications of Kronecker and Hadamard Products of Matrices
Mah'd Al Zhour, Zeyad Abdel Aziz (2006) Generalizations and Some Applications of Kronecker and Hadamard Products of Matrices. PhD thesis, Universiti Putra Malaysia.
In this thesis, generalizations of Kronecker, Hadamard and usual products (sums) that depend on the partitioned of matrices are studied and defined. Namely: Tracy- Singh, Khatri-Rao, box, strong Kronecker, block Kronecker, block Hadamard, restricted Khatri-Rao products (sums) which are extended the meaning of Kronecker, Hadamard and usual products (sums). The matrix convolution products, namely: matrix convolution, Kronecker convolution and Hadamard convolution products of matrices with entries in set of functions are also considered. The connections among them are derived and most useful properties are studied in order to find new applications of Tracy-Singh and Khatri-Rao products (sums). These applications are: a family of generalized inverses, a family of coupled singular matrix problems, a family of matrix inequalities and a family of geometric means. In the theory of generalized inverses of matrices and their applications, the five generalized inverses, namely Moore-Penrose, weighted Moore-Penrose, Drazin, weighted Drazin and group inverses and their expressions and properties are studied. Moreover, some new numerous matrix expressions involving these generalized inverses and weighted matrix norms of the Tracy-Singh products matrices are also derived. In addition, we establish some necessary and sufficient conditions for the reverse order law of Drazin and weighted Drazin inverses. These results play a central role in our applications and many other applications. In the field of system identification and matrix products work, we propose several algorithms for computing the solutions of the coupled matrix differential equations, coupled matrix convolution differential, coupled matrix equations, restricted coupled singular matrix equations, coupled matrix least-squares problems and weighted Least -squares problems based on idea of Kronecker (Hadamard) and Tracy-Singh(Khatri-Rao) products (sums) of matrices. The way exists which transform the coupled matrix problems and coupled matrix differential equations into forms for which solutions may be readily computed. The common vector exact solutions of these coupled are presented and, subsequently, construct a computationally - efficient solution of coupled matrix linear least-squares problems and nonhomogeneous coupled matrix differential equations. We give new applications for the representations of weighted Drazin, Drazin and Moore-Penrose inverses of Kronecker products to the solutions of restricted singular matrix and coupled matrix equations. The analysis indicates that the Kronecker (Hadamard) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal. Several special cases of these systems are also considered and solved, and then we prove the existence and uniqueness of the solution of each case, which includes the well-known coupled Sylvester matrix equations. We show also that the solutions of non-homogeneous matrix differential equations can be written in convolution forms. The analysis indicates also that the algorithms can be easily to find the common exact solutions to the coupled matrix and matrix differential equations for partitioned matrices by using the connections between Tracy-Singh, Block Kronecker and Khatri -Rao products and partitioned vector row (column) and our definition which is the so-called partitioned diagonal extraction operators. Unlike Matrix algebra, which is based on matrices, analysis must deal with estimates. In other words, Inequalities lie at the core of analysis. For this reason, it’s of great importance to give bounds and inequalities involving matrices. In this situation, the results are organized in the following five ways: First, we find some extensions and generalizations of the inequalities involving Khatri-Rao products of positive (semi) definite matrices. We turn to results relating Khatri-Rao and Tracy- Singh powers and usual powers, extending and generalizing work of previous authors. Second, we derive some new attractive inequalities involving Khatri-Rao products of positive (semi) definite matrices. We remark that some known inequalities and many other new interesting inequalities can easily be found by using our approaches. Third, we study some sufficient and necessary conditions under which inequalities below become equalities. Fourth, some counter examples are considered to show that some inequalities do not hold in general case. Fifth, we find Hölder-type inequalities for Tracy-Singh and Khatri-Rao products of positive (semi) definite matrices. The results lead to inequalities involving Hadamard and Kronecker products, as a special case, which includes the well-known inequalities involving Hadamard product of matrices, for instance, Kantorovich-type inequalities and generalization of Styan's inequality. We utilize the commutativity of the Hadamard product (sum) for possible to develop and improve some interesting inequalities which do not follow simply from the work of researchers, for example, Visick's inequality. Finally, a family of geometric means for positive two definite matrices is studied; we discuss possible definitions of the geometric means of positive definite matrices. We study the geometric means of two positive definite matrices to arrive the definitions of the weighted operator means of positive definite matrices. By means of several examples, we show that there is no known definition which is completely satisfactory. We have succeeded to find many new desirable properties and connections for geometric means related to Tracy-Singh products in order to obtain new unusual estimates for the Khatri-Rao (Tracy-Singh) products of several positive definite matrices.
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