Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Oppenheim and Schur Type Inequalities for Khatri-Rao Products of Positive Definite Matrices

  • Kim, Sejong (Department of Mathematics, Chungbuk National University) ;
  • Kim, Jungjoon (School of Electronics Engineering, Kyungpook National University) ;
  • Lee, Hosoo (Department of Mathematics, Sungkyunkwan University)
  • Received : 2017.09.01
  • Accepted : 2017.11.27
  • Published : 2017.12.23
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Abstract

For partitioned matrices, the Khatri-Rao product is viewed as a generalized Hadamard product. In this paper we present Oppenheim's and Schur's determinantal inequalities for the Khatri-Rao product of two positive semidefinite matrices.

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References

  1. S. S. Agaian, Hadamard Matrices and Their Applications, Springer-Verlag, Berlin, 1985.
  2. R. Horn and C. Johnson, Matrix analysis, 2nd edition, Cambridge University Press, Cambridge, 2013.
  3. A. Hedayat, W. D. Wallis, Hadamard matrices and their applications, Annals of Statistics, 6(1978), 11841238.
  4. C. G. Khatri and C. R. Rao, Solutions to some functional equations and their applications to charaterization of probability distributions, Sankhya, 30(1968), 167-180.
  5. M. Lin, An Oppenheim type inequality for a block Hadamard product, Linear Algebra Appl., 452(2014), 1-6. https://doi.org/10.1016/j.laa.2014.03.025
  6. S. Liu, Contributions to Matrix Calculus and Applications in Econometrics, Thesis Publishers, Amsterdam, The Netherlands, 1995.
  7. S. Liu, Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl., 289(1999), 267-277. https://doi.org/10.1016/S0024-3795(98)10209-4
  8. C. R. Rao, Estimation of heteroscedastic variances in linear models, J. Amer. Statist. Assoc., 65(1970), 161-172. https://doi.org/10.1080/01621459.1970.10481070
  9. C. R. Rao and J. Kleffe, Estimation of variance components and applications, North-Holland, Amsterdam, The Netherlands, 1988.
  10. G. P. H. Styan, Hadamard products and multivariate statistical analysis, Linear Algebra Appl., 6(1973), 217-240. https://doi.org/10.1016/0024-3795(73)90023-2
  11. J. Seberry, M. Yamada, Hadamard matrices, sequences, and block designs, in : J. H. Dinitz, D. R. Stinson(Eds.), Contemporary Design Theory : A Collection of Surveys, John Wiley Sons, NewYork, 1992 (Chapter11).
  12. K. Vijayan, Hadamard matrices and submatrices, J. Austr. Math. Soc. Ser. A, 22(1976), 469-475. https://doi.org/10.1017/S1446788700016335