A matrix sequence {Gamma(A(m))}(m=1)(infinity) might converge even if the matrix A is not primitive

作者：Park Woongbae; Park Boram^*; Kim Suh Ryung

来源：Linear Algebra and Its Applications, 2013, 438(5): 2306-2319.

DOI：10.1016/j.laa.2012.10.012

• Irreducible Boolean (0,1)-matrix
• Powers of Boolean (0,1)-matrices
• Competition graph
• Graph sequence
• Powers of digraphs

It is well-known that, for an irreducible Boolean (0, 1)-matrix A, the matrix sequence {A(m)}(m=1)(infinity) converges if and only if A is primitive. In this paper, we introduce an operation Gamma on the set of Boolean (0, 1)-matrices such that a matrix sequence {Gamma(A(m))}(m=1)(infinity) might converge even if the matrix A is not primitive. Given a Boolean (0, 1)-matrix A, we define a matrix Gamma (A) so that the (i, j)-entry of Gamma(A) equals 0 if for i not equal j, the inner product of the ith row and jth row of A is 0 and equals 1 otherwise.
The aim of this paper is to study the convergence of {Gamma(A(m))}(m=1)(infinity) for a Boolean (0, 1)-matrix A whose digraph has at most two strong components. We show that {Gamma(A(m))}(m=1)(infinity) converges to a very special type of matrix as m increases if A is an irreducible Boolean matrix. Furthermore, we completely characterize a Boolean (0, 1)-matrix A whose digraph has exactly two strongly connected components and for which {Gamma(A(m))}(m=1)(infinity) converges, and find the limit of {Gamma(A(m))}(m=1)(infinity) in terms of its digraph when it converges. We derive these results in terms of the competition graph of the digraph of A.

• 出版日期2013-3-1