tailieunhanh - Báo cáo hóa học: "EXTENSION AND GENERALIZATION INEQUALITIES INVOLVING THE KHATRI-RAO PRODUCT OF SEVERAL POSITIVE MATRICE"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: EXTENSION AND GENERALIZATION INEQUALITIES INVOLVING THE KHATRI-RAO PRODUCT OF SEVERAL POSITIVE MATRICE | EXTENSION AND GENERALIZATION INEQUALITIES INVOLVING THE KHATRI-RAO PRODUCT OF SEVERAL POSITIVE MATRICES ZEYAD ABDEL AZIZ AL ZHOUR AND ADEM KILICMAN Received 15 February 2005 Accepted 16 October 2005 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. Copyright 2006 Z. A. Al Zhour and A. Kilicman. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Consider matrices A and B of order m X n and p X q respectively. Let A Aij be partitioned with Aij of order mị X nj as the i j th block submatrix and let B Bki be partitioned with Bki of order pk X qi as the k l th block submatrix m - 1 mi n X j i nj p sk 1 pk q sv 1 qi . For simplicity we say that A and B are compatible partitioned if A Aij - j 1 and B Bij j 1 are square matrices of order m X m and partitioned respectively with Aij and Bij of order mi X mj m - 1 mi sj 1 mj . Let A B A B A B and A B be the Kronecker Hadamard Tracy-Singh and Khatri-Rao products respectively of A and B. The definitions of the mentioned four matrix products are given by Liu in 5 6 as follows i Kronecker product A B ỊữijB ịj where A aij B bki are scalar matrices of order m X n and p X q respectively aijB is of order p X q and A B of order mp X nq Hindawi Publishing Corporation Journal .