Firstly, the relationships among strictly diagonally dominant (
) matrices, doubly strictly diagonally dominant (
) matrices, eventually
matrices and eventually
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
matrices is derived.
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