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

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.

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References

  1. 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.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. 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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. 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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Zeyad Abdel Aziz Al Zhour.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Al Zhour, Z.A.A., Kilicman, A. Extension and generalization inequalities involving the Khatri-Rao product of several positive matrices. J Inequal Appl 2006, 80878 (2006). https://doi.org/10.1155/JIA/2006/80878

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  • DOI: https://doi.org/10.1155/JIA/2006/80878

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