Abstract
Haynsworth (Proc Am Math Soc 24:512–516, 1970) used a result of the Schur complement to refine a determinant inequality for positive definite matrices. Haynsworth’s result was improved by Hartfiel (Proc Am Math Soc 41:463–465, 1973). We extend their results to a larger class of matrices, namely, matrices whose numerical range is contained in a sector. Our proof relies on a number of new relations for the Schur complement of this class of matrices.
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References
Drury S. W.: Fischer determinantal inequalities and Higham’s Conjecture, Linear Algebra Appl.439, 3129–3133 (2013)
George A., Ikramov Kh. D., Kucherov A. B.: On the growth factor in Gaussian elimination for generalized Higham matrices, Numer. Linear Algebra Appl. 9, 107–114 (2002)
George A., Ikramov Kh. D.: On the properties of accretive-dissipative matrices, Math. Notes 77, 767–776 (2005)
K. E. Gustafson and D. K. M. Rao , Numerical Range: The Field of Values of Linear Operators and Matrices, Springer, New York, 1997.
Hartfiel D. J.: An extension of Haynsworth’s determinant inequality, Proc. Amer. Math. Soc. 41, 463–465 (1973)
Haynsworth E. V.: Applications of an inequality for the Schur complement, Proc. Amer. Math. Soc. 24, 512–516 (1970)
Higham N. J.: Factorizing complex symmetric matrices with positive real and imaginary parts, Math. Comp. 67, 1591–1599 (1998)
R. A. Horn and C. R. Johnson , Topics in Matrix Analysis, Cambridge University Press, 1991.
R. A. Horn and C. R. Johnson , Matrix Analysis, Cambridge University Press, 2nd ed. 2013.
Li C.-K., Sze N.: Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector, J. Math. Anal. Appl. 410, 487–491 (2014)
Lin M.: Reversed determinantal inequalities for accretive-dissipative matrices, Math. Inequal. Appl. 12, 955–958 (2012)
Lin M.: A note on the growth factor in Gaussian elimination for accretive-dissipative matrices. Calcolo 51, 363–366 (2014)
Mathias R.: Matrices with positive definite Hermitian part: Inequalities and linear systems, SIAM J. Matrix Anal. Appl. 13, 640–654 (1992)
F. Zhang , The Schur complement and its applications, Springer, New York, 2005.
F. Zhang , A matrix decomposition and its applications, Linear Multilinear Algebra (2014) to appear. doi:10.1080/03081087.2014.933219
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Lin, M. Extension of a result of Haynsworth and Hartfiel. Arch. Math. 104, 93–100 (2015). https://doi.org/10.1007/s00013-014-0717-2
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DOI: https://doi.org/10.1007/s00013-014-0717-2