迹非零几乎可约分块布尔矩阵的幂敛指数

The Index of Convergence of Nearly Reducible Block Matrices

中文关键词:  布尔矩阵  幂敛指数  上确界  极矩阵.

英文关键词:Boolean matrix  convergent index  exact upper bound  extreme matrix.

中文摘要:

      设$H_n (d)$ 是恰含 $d$个正对角元的$n$阶几乎可约分块布尔矩阵的集合, $1 \le d \le n$,对任何矩阵$A \in H_n (d)$,本文证明了\[k(A) \le \left\{ {\begin{array}{ll} (n - d - 2)^2 + 2 , & 1 \le d \le s_n \\ 2n - d - 1, & s_n \le d_n \le n \end{array}} \right.\]其中$s_n = \left\lfloor {\frac{2n - 5 - \sqrt {4n - 3} }{2}}\right\rfloor $,同时刻画了$H_n (d)$中幂敛指数达到最大值的极矩阵.

英文摘要:

      Let $H_n(d)$, $1\le d\le n$, be the set of nearly reducible Boolean block matrices of order $n$ with exact $d$ non--zero diagnols. The index of convergence of a matrix $A$ is denoted by $k(A)$. This paper solves the problem for the exact upper bound of $k(A)$ completely. The following result is proved: $$\align k\left( {v_i ,v_j } \right) &\le \max \left\{ {\left( {n - d - 2}\right)^2 + 2,2n - d - 1} \right\}\\&\le \left\{ {\begin{array}{l}(n - d - 2)^2 + 2, \quad\quad 1 \le d \le s_n \\2n - d - 1,\quad\quad\quad\quad s_n < d \le n\end{array} }\right.\endalign$$$$s_n = \left\lfloor {\frac{(2n - 5) - \sqrt {4n - 3} }{2}}\right\rfloor.$$And we give complete characterization for the extreme matrices with the largest convergent index in $H_n(d)$.