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

Affiliation(s)

Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China.

Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal.

Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China.

Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal.

ABSTRACT

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

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.

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,"

References

[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.

[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.