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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2006 | Research Article

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

verfasst von: Zeyad Abdel Aziz Al Zhour, Adem Kilicman

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2006

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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 Pečarić (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.

Vorheriger Artikel Existence and multiplicity of solutions for some three-point nonlinear boundary value problems

Nächster Artikel On random coincidence and fixed points for a pair of multivalued and single-valued mappings

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Al Zhour ZA, Kilicman A: New Holder-type inequalities for the Tracy-Singh and Khatri-Rao products of positive matrices. Proceedings of the International Conference on Mathematics, Statistics and Their Applications, 2005, North Sumatera 1: 1–7.

Albert A: Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM Journal on Applied Mathematics 1969,17(2):434–440. 10.1137/0117041MATHMathSciNetCrossRef

Cao C-G, Zhang X, Yang Z-P: Some inequalities for the Khatri-Rao product of matrices. Electronic Journal of Linear Algebra 2002, 9: 276–281.MATHMathSciNetCrossRef

Chollet J: Some inequalities for principal submatrices. The American Mathematical Monthly 1997,104(7):609–617. 10.2307/2975054MATHMathSciNetCrossRef

Liu S: Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra and Its Applications 1999,289(1–3):267–277.MATHMathSciNetCrossRef

Liu S: Several inequalities involving Khatri-Rao products of positive semidefinite matrices. Linear Algebra and Its Applications 2002,354(1–3):175–186.MATHMathSciNetCrossRef

Mond B, Pečarić JE: Matrix inequalities for convex functions. Journal of Mathematical Analysis and Applications 1997,209(1):147–153. 10.1006/jmaa.1997.5353MATHMathSciNetCrossRef

Visick G: A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product. Linear Algebra and Its Applications 2000,304(1–3):45–68.MATHMathSciNetCrossRef

Zhang F: Matrix Theory. Basic Results and Techniques, Universitext. Springer, New York; 1999:xiv+277.

10.

Zhang X, Yang Z-P, Cao C-G: Inequalities involving Khatri-Rao products of positive semi-definite matrices. Applied Mathematics E-Notes 2002, 2: 117–124.MATHMathSciNet

Titel: Extension and generalization inequalities involving the Khatri-Rao product of several positive matrices
verfasst von: Zeyad Abdel Aziz Al Zhour
Adem Kilicman
Publikationsdatum: 01.12.2006
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2006
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1155/JIA/2006/80878

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