Aequationes Mathematicae
نویسنده
چکیده
Let A be an M-matrix in standard lower block triangular form, with diagonal blocks Ai; irreducible. Let S be the set of indices a such that the diagonal block A"" is singular. We define the singular graph of A to be the set S with partial order defined by a > (3 if there exists a chain of non-zero blocks A,,;, A;i' .. . , AIi3" Let Al be the set of maximal elements of S, and define the p-th level Ap' p = 2, 3, ... , inductively as the set of maximal elements of S\(A I U . .. U Ap_I ). Denote by Ap the number of elements in Ap. The Weyr characteristic (associated with 0) of A is defined to be w(A ) = (WI' W2" •• , wh ) , where WI + ... + w p = dim Ker AP, p = 1,2, ... , and W. > 0, W.+I = O. Using a special type of basis, called an S-basis, for the generalized eigenspace E(A) of 0 of A, we associate a matrix D with A. We show that w(A) = (AI' ... , Ah) if and only if certain submatrices Dp•p + t ' P = 1, ... , h -1, of D have full column rank. This condition is also necessary and sufficient for E(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for -A. We also consider a given finite partially ordered set S, and we find a necessary and sufficient condition that all M-matrices A with singular graph Shave w(A) = (AI" .. ,A.). This condition is satisfied if S is a rooted forest.
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تاریخ انتشار 2009