## The dominant degree and disc theorem for the Schur complement of matrix.(English)Zbl 1189.15023

Authors’ abstract: The theory of Schur complement plays an important role in many fields such as control theory and computational mathematics. In this paper, we obtain some estimates for the diagonally, $$\gamma$$-diagonally and product $$\gamma$$-diagonally dominant degree on the Schur complement of matrices, which improve some relative results. As application we present some bounds for the eigenvalues of Schur complement by the entries of the original matrix instead of those of the Schur complement. Particularly, we obtain that the eigenvalues of the Schur complements are located in the Gerschgorin circles of the original matrices under certain conditions.

### MSC:

 15A42 Inequalities involving eigenvalues and eigenvectors
Full Text:

### References:

 [1] Johnson, C.R., Inverse M-matrices, Linear algebra appl., 47, 195-216, (1982) · Zbl 0488.15011 [2] Carlson, D.; Markham, T., Schur complements on diagonally dominant matrices, Czech. math. J., 29, 104, 246-251, (1979) · Zbl 0423.15008 [3] Li, B.; Tsatsomeros, M., Doubly diagonally domiant matrices, Linear algebra appl., 261, 221-235, (1997) · Zbl 0886.15027 [4] Ikramov, K.D., Invariance of the Brauer diagonal dominance in Gaussian elimination, Moscow univ. comput. math. cybernet. $$(N_2)$$, 91-94, (1989) · Zbl 0723.65013 [5] Liu, J.Z.; Huang, Y.Q., The Schur comeplements of generalized doubly diagonlly dominant matrices, Linear algebra appl., 378, 231-244, (2004) [6] Liu, J.Z.; Huang, Y.Q., Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear algebra appl., 389, 365-380, (2004) · Zbl 1068.15004 [7] J.Z. Liu and F.Z. Zhang, Disc separation of the Schur complements of diagonally dominant matrices and determinantal bounds, SIAM J. Matrix Anal. Appl. 2(3) 665-674. · Zbl 1107.15022 [8] Liu, J.Z., Some inequalities for singular values and eigenvalues of generalized Schur complements of products of matrices, Linear algebra appl., 286, 209-221, (1999) · Zbl 0941.15017 [9] Liu, J.Z.; Zhu, L., A minimum principle and estimates of the eigenvalues for Schur complements of positive semidefinite Hermitian matrices, Linear algebra appl., 265, 123-145, (1997) · Zbl 0885.15010 [10] Smith, R., Some interlacing properties of the Schur complement of a Hermitian matrix, Linear algebra appl., 177, 137-144, (1992) · Zbl 0765.15007 [11] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), Johns Hopkins University Press Baltimore · Zbl 0865.65009 [12] Kress, R., Numerical analysis, (1998), Springer New York [13] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press New York · Zbl 0729.15001 [14] Liu, J.Z., Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear algebra appl., 428, 1009-1030, (2008) · Zbl 1133.15020 [15] Ostrowski, A., Uber das nichverchwinder einer klass von determinanten und die lokalisierung der charakterischen wurzeln von matrizen, Composition math., 9, 209-226, (1951) · Zbl 0043.01703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.