A New Upper Bound for of a Strictly -Diagonally Dominant -Matrix

A new upper bound for of a real strictly diagonally dominant -matrix is present, and a new lower bound of the smallest eigenvalue of is given, which improved the results in the literature. Furthermore, an upper bound for of a real strictly -diagonally dominant -matrix is shown. 1. Introduction The estimation for the bound for the norm of a real invertible matrix is important in numerical analysis, so many researchers were devoted to studying this kind of problems. For example, Varah [1] discussed the bound for the infinity norm of a strictly diagonally dominant matrix and obtained the following estimation: After that Varga [2] extended the result of [1] to -matrices. Evidently, the upper bound for in (1) only involves the entries in the matrix . If the diagonal dominance of is weak, that is, is small, then the bound given by (1) may be large. For this reason, some authors were devoted to improving the result of (1). Recently, Cheng and Huang [3] presented a more compacted upper bound for a strictly diagonally dominant -matrix and then Wang [4] further improved this bound and gave the following result: where notations in (2) and (3) have the same meanings as those used in this paper, which will be shown later. In this paper, we present a new upper bound of a strictly diagonally dominant matrix , which is better than that obtained by Wang, and a new lower bound of the smallest eigenvalue of is also obtained. In addition, an upper bound for of a strictly -diagonal dominant matrix is presented. To our knowledge, little has been done for upper bound of strictly -diagonal dominant matrices. Further, examples are given to illustrate the performance of our results. Next, we introduce some notations and definitions. As usual, let be an identity matrix of order . If there exists an nonnegative matrix and a real number such that with , then is called a nonsingular -matrix, where is the spectral radius of the nonnegative matrix . It is well known that the inverse matrix of a -matrix is nonnegative and, therefore, is a positive eigenvalue of related to the Perron eigenvalue of the nonnegative matrix . If denotes the minimum of the real parts of the eigenvalues of , that is, , then . For further properties of the -matrix , we refer the readers to [5–7]. An matrix is called a strictly diagonally dominant matrix if for . Let where is the set of positive integers. For an matrix , the principal matrix of formed by rows and columns with indices between and is denoted by . Definition 1 (see [8]). is weakly chained diagonally dominant if, for all , and and for all , , there