THE GENERALIZATION OF STYAN MATRIX INEQUALITY ON HERMITIAN MATRICES

  • Published : 2009.05.31

Abstract

We point out: to make Hermtian matrices A and B satisfy Styan matrix inequality, the condition "positive definite property" demanded in the present literatures is not necessary. Furthermore, on the premise of abandoning positive definite property, we derive Styan matrix inequality of Hadamard product for inverse Hermitian matrices and the sufficient and necessary conditions that the equation holds in our paper.

Keywords

References

  1. T. Ando, Concavity of certain maps on positive definite matrices and applications for Hadamard products, Linear Algebra Appl., 26(1979), 203-241. https://doi.org/10.1016/0024-3795(79)90179-4
  2. R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge U. P, Cambridge U. K, 1991.
  3. R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University press, New York, 1985.
  4. S. Liu, Note inequalities involving Hadamard products of positive semidefinite matrices, J. Math. Anal. Appl., 243(2000), 458-463. https://doi.org/10.1006/jmaa.1999.6670
  5. S. Liu, Several inequalities involving Khatri-Rao products of positive semidefinite matri- ces, Linear Algebra Appl., 354(2002), 175-186.
  6. S. Liu, W. Polasck and H. Neudecker, Equality conditions for matrix Kantorovich-type inequalities, J. Math. Anal. Appl., 212(1997), 517-528. https://doi.org/10.1006/jmaa.1997.5526
  7. T. L. Markham and R. L. Smith, A Schur complement inequality for certain P- matrices, Linear Algebra Appl., 281(1998), 33-41. https://doi.org/10.1016/S0024-3795(98)10023-X
  8. 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
  9. G. Visick, A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product, Linear Algebra Appl., 304(2000), 45-68. https://doi.org/10.1016/S0024-3795(99)00187-1
  10. Z. Yang, C. Cao and X. Zhang, A Matrix inequality on Shur complements, Journal of Applied Mathematics and Computing, 15(1-2)(2005), 321-328.
  11. F. Zhang, Schur complements and matrix inequalities in the Lowner ordering, Linear Algebra Appl., 321(2000), 399-410. https://doi.org/10.1016/S0024-3795(00)00032-X
  12. F. Zhang, The Schur complements and its applications, Numerical methods and algorithms, vol 4, New York: Springer, 2005.
  13. F. Zhang, A matrix identity on the schur complements, Linear and Multinear Algebra, 52(5)(2004), 367-373.
  14. Z. A.Al Zhour and A.Kilicman, Extension and generalization inequalities involving the Khatri-Rao product of several positivematrices, J. Inequal. Appl., (2006a), Art. ID 80878.
  15. Z. A. Al Zhour and A. Kilicman, Matrix equalities and inequalities in- volving Khatri-Rao and Tracy-Singh sums, J. Inequalities Pure Appl. Math., 7(1)(Article 34)(2006), 1-17.