The Estimates of Diagonally Dominant Degree and Eigenvalue Inclusion Regions for the Schur Complement of Matrices
Author(s)
Dongjie Gao*
ABSTRACT
The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.
The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.
Cite this paper
Gao, D. (2015) The Estimates of Diagonally Dominant Degree and Eigenvalue Inclusion Regions for the Schur Complement of Matrices. Advances in Pure Mathematics, 5, 643-652. doi: 10.4236/apm.2015.510058.
Gao, D. (2015) The Estimates of Diagonally Dominant Degree and Eigenvalue Inclusion Regions for the Schur Complement of Matrices. Advances in Pure Mathematics, 5, 643-652. doi: 10.4236/apm.2015.510058.
References
[1] Cvetkovic, Lj. (2009) A New Subclass of h-Matrices. Applied Mathematics and Computation, 208, 206-210.
http://dx.doi.org/10.1016/j.amc.2008.11.037 [2] Liu, J.Z. and Huang, Z.J. (2010) The Dominant Degree and Disc Theorem for the Schur Complement. Applied Mathematics and Computation, 215, 4055-4066.
http://dx.doi.org/10.1016/j.amc.2009.12.063 [3] Horn, R.A. and Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9780511840371 [4] Carlson, D. and Markham, T. (1979) Schur Complements on Diagonally Dominant Matrices. Czechoslovak Mathematical Journal, 29, 246-251. [5] Ikramov, K.D. (1989) Invariance of the Brauer Diagonal Dominance in Gaussian Elimination. Moscow University Computational Mathematics and Cybernetics, 2, 91-94. [6] Li, B. and Tsatsomeros, M. (1997) Doubly Diagonally Dominant Matrices. Linear Algebra and Its Applications, 261, 221-235.
http://dx.doi.org/10.1016/S0024-3795(96)00406-5 [7] Smith, R. (1992) Some Interlacing Properties of the Schur Complement of a Hermitian Matrix. Linear Algebra and Its Applications, 177, 137-144.
http://dx.doi.org/10.1016/0024-3795(92)90321-Z [8] Zhang, F.Z. (2005) The Schur Complement and Its Applications. Springer-Verlag, New York.
http://dx.doi.org/10.1007/b105056 [9] Demmel, J.W. (1997) Applied Numerical Linear Algebra. SIAM, Philadephia. [10] Golub, G.H. and Van Loan, C.F. (1996) Matrix Computations. 3rd Edition, Johns Hopkins University Press, Baltimore. [11] Kress, R. (1998) Numerical Analysis. Springer, New York.
http://dx.doi.org/10.1007/978-1-4612-0599-9 [12] Xiang, S.H. and Zhang, S.L. (2006) A Convergence Analysis of Block Accelerated Over-Relaxation Iterative Methods for Weak Block H-Matrices to Partition π. Linear Algebra and Its Applications, 418, 20-32.
http://dx.doi.org/10.1016/j.laa.2006.01.013 [13] Liu, J.Z., Li, J.C., Huang, Z.H. and Kong, X. (2008) Some Properties on Schur Complement and Diagonal Schur Complement of Some Diagonally Dominant Matrices. Linear Algebra and Its Applications, 428, 1009-1030.
http://dx.doi.org/10.1016/j.laa.2007.09.008 [14] Liu, J.Z. and Huang, Y.Q. (2004) The Schur Complements of Generalized Doubly Diagonally Dominant Matrices. Linear Algebra and Its Applications, 378, 231-244.
http://dx.doi.org/10.1016/j.laa.2003.09.012 [15] Liu, J.Z. and Huang, Y.Q. (2004) Some Properties on Schur Complements of H-Matrices and Diagonally Dominant Matrices. Linear Algebra and Its Applications, 389, 365-380.
http://dx.doi.org/10.1016/j.laa.2004.04.012 [16] Liu, J.Z. and Huang, Z.J. (2010) The Schur Complements of γ-Diagonally and Product γ-Diagonally Dominant Matrix and their Disc Separation. Linear Algebra and Its Applications, 432, 1090-1104.
http://dx.doi.org/10.1016/j.laa.2009.10.021 [17] Liu, J.Z. and Zhang, F.Z. (2005) Disc Separation of the Schur Complements of Diagonally Dominant Matrices and Determinantal Bounds. SIAM Journal on Matrix Analysis and Applications, 27, 665-674.
http://dx.doi.org/10.1137/040620369 [18] Li, Y.T. Ouyang, S.P. Cao, S.J. and Wang, R.W. (2010) On Diagonal-Schur Complements of Block Diagonally Dominant Matrices. Applied Mathematics and Computation, 216, 1383-1392.
http://dx.doi.org/10.1016/j.amc.2010.02.038
[1] Cvetkovic, Lj. (2009) A New Subclass of h-Matrices. Applied Mathematics and Computation, 208, 206-210.
http://dx.doi.org/10.1016/j.amc.2008.11.037 [2] Liu, J.Z. and Huang, Z.J. (2010) The Dominant Degree and Disc Theorem for the Schur Complement. Applied Mathematics and Computation, 215, 4055-4066.
http://dx.doi.org/10.1016/j.amc.2009.12.063 [3] Horn, R.A. and Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9780511840371 [4] Carlson, D. and Markham, T. (1979) Schur Complements on Diagonally Dominant Matrices. Czechoslovak Mathematical Journal, 29, 246-251. [5] Ikramov, K.D. (1989) Invariance of the Brauer Diagonal Dominance in Gaussian Elimination. Moscow University Computational Mathematics and Cybernetics, 2, 91-94. [6] Li, B. and Tsatsomeros, M. (1997) Doubly Diagonally Dominant Matrices. Linear Algebra and Its Applications, 261, 221-235.
http://dx.doi.org/10.1016/S0024-3795(96)00406-5 [7] Smith, R. (1992) Some Interlacing Properties of the Schur Complement of a Hermitian Matrix. Linear Algebra and Its Applications, 177, 137-144.
http://dx.doi.org/10.1016/0024-3795(92)90321-Z [8] Zhang, F.Z. (2005) The Schur Complement and Its Applications. Springer-Verlag, New York.
http://dx.doi.org/10.1007/b105056 [9] Demmel, J.W. (1997) Applied Numerical Linear Algebra. SIAM, Philadephia. [10] Golub, G.H. and Van Loan, C.F. (1996) Matrix Computations. 3rd Edition, Johns Hopkins University Press, Baltimore. [11] Kress, R. (1998) Numerical Analysis. Springer, New York.
http://dx.doi.org/10.1007/978-1-4612-0599-9 [12] Xiang, S.H. and Zhang, S.L. (2006) A Convergence Analysis of Block Accelerated Over-Relaxation Iterative Methods for Weak Block H-Matrices to Partition π. Linear Algebra and Its Applications, 418, 20-32.
http://dx.doi.org/10.1016/j.laa.2006.01.013 [13] Liu, J.Z., Li, J.C., Huang, Z.H. and Kong, X. (2008) Some Properties on Schur Complement and Diagonal Schur Complement of Some Diagonally Dominant Matrices. Linear Algebra and Its Applications, 428, 1009-1030.
http://dx.doi.org/10.1016/j.laa.2007.09.008 [14] Liu, J.Z. and Huang, Y.Q. (2004) The Schur Complements of Generalized Doubly Diagonally Dominant Matrices. Linear Algebra and Its Applications, 378, 231-244.
http://dx.doi.org/10.1016/j.laa.2003.09.012 [15] Liu, J.Z. and Huang, Y.Q. (2004) Some Properties on Schur Complements of H-Matrices and Diagonally Dominant Matrices. Linear Algebra and Its Applications, 389, 365-380.
http://dx.doi.org/10.1016/j.laa.2004.04.012 [16] Liu, J.Z. and Huang, Z.J. (2010) The Schur Complements of γ-Diagonally and Product γ-Diagonally Dominant Matrix and their Disc Separation. Linear Algebra and Its Applications, 432, 1090-1104.
http://dx.doi.org/10.1016/j.laa.2009.10.021 [17] Liu, J.Z. and Zhang, F.Z. (2005) Disc Separation of the Schur Complements of Diagonally Dominant Matrices and Determinantal Bounds. SIAM Journal on Matrix Analysis and Applications, 27, 665-674.
http://dx.doi.org/10.1137/040620369 [18] Li, Y.T. Ouyang, S.P. Cao, S.J. and Wang, R.W. (2010) On Diagonal-Schur Complements of Block Diagonally Dominant Matrices. Applied Mathematics and Computation, 216, 1383-1392.
http://dx.doi.org/10.1016/j.amc.2010.02.038