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**A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix** ()

We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.

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W. Mai, M. Yan, T. Qian, M. Riva and S. Saitoh, "A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix,"

*Advances in Linear Algebra & Matrix Theory*, Vol. 3 No. 4, 2013, pp. 55-58. doi: 10.4236/alamt.2013.34011.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | S. Saitoh, “Integral Transforms, Reproducing Kernels and Their Applications,” Pitman Research Notes in Mathematics Series 369, Addison Wesley Longman, Harlow, 1997. |

[2] | S. Saitoh, “Theory of Reproducing Kernels: Applications to Approximate Solutions of Bounded Linear Operator Functions on Hilbert Spaces,” American Mathematical Society Translations: Series 2, Vol. 230, American Mathematical Society, Providence, 2010. |

[3] | S. Saitoh, “The Bergman Norm and the Szeg? Norm,” Transactions of the American Mathematical Society, Vol. 249, No. 1-2, 1979, pp. 261-279. |

[4] | M. Asaduzzaman and S. Saitoh, “Inverses of a Family of Matrices and Generalizations of Pythagorean Theorem,” Panamerican Mathematical Journal, Vol. 13, No. 4, 2003, pp. 45-53. |

[5] | B. Mond, J. E. Pecaric and S. Saitoh, “History, Variations and Generalizations of an Inequality of Marcus,” Riazi. The Journal of Karachi Mathematical Association, Vol. 16, No. 1, 1994, pp. 7-15. |

[6] | S. Saitoh, “Positive Definite Hermitian Matrices and Reproducing Kernels,” Linear Algebra and Its Applications, Vol. 48, No. 1, 1982, pp. 119-130. |

[7] | S. Saitoh, “Quadratic Inequalities Deduced from the Theory of Reproducing Kernels,” Linear Algebra and Its Applications, Vol. 93, No. 1, 1987, pp. 171-178. |

[8] | S. Saitoh, “Quadratic Inequalities Associated with Integrals of Reproducing Kernels,” Linear Algebra and Its Applications, Vol. 101, No. 2, 1988, pp. 269-280. |

[9] | S. Saitoh, “Generalizations of the Triangle Inequality,” JIPAM—Journal of Inequalities in Pure and Applied Mathematics, Vol. 4, No. 3, 2003, Article 62. |

[10] | Y. Sawano, “Pasting Reproducing Kernel Hilbert Spaces,” Jaen Journal on Approximation, Vol. 3, No. 1, 2011, pp. 135-141. |

[11] | A. Yamada, “Oppenheim’s Inequality and RKHS,” Mathematical Inequalities & Applications, Vol. 15, No. 2, 2012, pp. 449-456. |

[12] | A. Yamada, “Inequalities for Gram Matrices and Their Applications to Reproducing Kernel Hilbert Spaces,” Taiwanese Journal of Mathematics, Vol. 17, No. 2, 2013, pp. 427-430. |

[13] | D. Carlson, “What Are Schur Complements, Anyway?” Linear Algebra and Its Applications, Vol. 74, No. 1, 1986, pp. 257-275. |

[14] | L. P. Castro, H. Fujiwara, M. M. Rodrigues, S. Saitoh and V. K. Tuan, “Aveiro Discretization Method in Mathematics: A New Discretization Principle,” Mathematics with out Boundaries: Surveys in Pure Mathematics, Edited by Panos Pardalos and Themistocles M. Rassias (to appear). 52 p. |

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