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

Zeyad Abdel Aziz Al Zhour,Adem Kilicman

doi:10.1155/jia/2006/80878

Zeyad Abdel Aziz Al Zhour, Adem Kilicman

Open Access

https://doi.org/10.1155/jia/2006/80878

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Abstract

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.

Highlights

Consider matrices A and B of order m × n and p × q, respectively
In [5,6,7,8], the authors proved a number of equalities and inequalities involving KhatriRao and Hadamard products of two matrices
The results are extended to the case of Khatri-Rao products of any finite number of matrices. This result leads to inequalities involving Hadamard product, as a special case

Summary

Introduction

Consider matrices A and B of order m × n and p × q, respectively. Let A = [Aij] be partitioned with Aij of order mi × nj as the (i, j)th block submatrix and let B = [Bkl] be partitioned with Bkl of order pk × ql as the (k, l)th block submatrix Let A ⊗ B, A ◦ B, AΘB, and A ∗ B be the Kronecker, Hadamard, Tracy-Singh, and Khatri-Rao products, respectively, of A and B. The Khatri-Rao and Tracy-Singh products are related by the following relation [5, 6]:. In [5,6,7,8], the authors proved a number of equalities and inequalities involving KhatriRao and Hadamard products of two matrices. We extend these results in three ways. The results are extended to the case of Khatri-Rao products of any finite number of matrices This result leads to inequalities involving Hadamard product, as a special case. We use the known fact “for positive definite matrices A and B, the relation A ≥ B implies A1/2 ≥ B1/2” which is called the Lowner-Heinz theorem

Some notations and preliminary results

New applications and results

Further developments and applications