The Index of Convergence of Nearly Reducible Block Matrices
Received:September 08, 2003  
Key Words: Boolean matrix   convergent index   exact upper bound   extreme matrix.  
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Author NameAffiliation
JIANG Zhi-ming Dept. of Math., East China University of Science and Technology, Shanghai 201512, China 
WANG Zong-yao Dept. of Math., East China University of Science and Technology, Shanghai 201512, China 
LIU Bo-lian Dept. of Math., South China Normal University, Guangzhou 510631, China 
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Abstract:
      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 $H_n(d)$.
Citation:
DOI:10.3770/j.issn:1000-341X.2006.03.031
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