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On Green's Matrices of Trees

The inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is a so-called Green's matrix. A Green's matrix is a symmetric matrix which is given by two sequences of real numbers {ui} and {vi} such that ci,j = uivj for $i \leq j$. A similar result holds for nonsymmetri...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 2001-01, Vol.22 (4), p.1014-1026
Main Author: Nabben, Reinhard
Format: Article
Language:English
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Summary:The inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is a so-called Green's matrix. A Green's matrix is a symmetric matrix which is given by two sequences of real numbers {ui} and {vi} such that ci,j = uivj for $i \leq j$. A similar result holds for nonsymmetric matrices. An open problem on nonsingular sparse matrices is whether there exists a similar structure for their inverses as in the tridiagonal case. Here we positively answer this question for irreducible acyclic matrices, i.e., matrices whose undirected graphs are trees. We prove that the inverses of irreducible acyclic symmetric matrices are given as the Hadamard product of three matrices, a type D matrix, a flipped type D matrix, and a matrix of tree structure which is closely related to the graph of the original matrix itself. For nonsymmetric matrices we obtain a similar structure. Moreover, our results include the result for symmetric and nonsymmetric tridiagonal matrices.
ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479899365732