Scientific Research

An Academic Publisher

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

**Author(s)**Leave a comment

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

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

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.

[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. |

Copyright © 2019 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.