Abstract
Firstly, the relationships among strictly diagonally dominant ( S D D ) matrices, doubly strictly diagonally dominant ( D S D D ) matrices, eventually S D D matrices and eventually D S D D matrices are considered. Secondly, by excluding some proper subsets of an existing eigenvalue inclusion set for matrices, which do not contain any eigenvalues of matrices, a tighter eigenvalue inclusion set of matrices is derived. As its application, a sufficient condition of determining non-singularity of matrices is obtained. Finally, the infinity norm estimation of the inverse of eventually D S D D matrices is derived.
Highlights
Let n be a positive integer, n ≥ 2, J = {1, 2, · · ·, n}, N be the set of all positive integers, C be the set of all complex numbers, Cn×n be the set of all n × n complex matrices and I be the identity matrix
The first problem is to find a set in the complex plane to include all eigenvalues of matrices, as if the obtained set is on the right-hand side of the complex plane
Min{|sk − ( Bk )ii | − ri ( Bk )}. It is worth noting in Theorem 1 that, from Examples 3 and 4 in [7] and the structure of Ψsk ( A), which includes two parameters s and k, one can conclude that, if there exists two groups of different numbers s and k, both such that A is an strictly diagonally dominant (SDD)∃ matrix, the different selection of s and k will affect the infinity norm bounds for the inverse of A
Summary
Locating eigenvalue and bounding infinity norm of the inverse for nonsingular matrices are two major problems in applied linear algebra [1,2,3,4,5,6,7]. It is worth noting in Theorem 1 that, from Examples 3 and 4 in [7] and the structure of Ψsk ( A), which includes two parameters s and k, one can conclude that, if there exists two groups of different numbers s and k, both such that A is an SDD∃ matrix, the different selection of s and k will affect the infinity norm bounds for the inverse of A. Besides SDD matrices, DSDD matrices, SDD∃ matrices and DSDD∃ matrices, there are many other subclasses of nonsingular matrices; see [11] for details.
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