Theorems on M-splittings of a singular M-matrix which depend on graph structure |
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Authors: | Hans Schneider |
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Institution: | Department of Mathematics University of Wisconsin—Madison Madison, Wisconsin 53706, USA |
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Abstract: | Let be a splitting. We investigate the spectral properties of the iteration matrix M-1N by considering the relationships of the graphs of A, M, N, and M-1N. We call a splitting an M-splitting if M is a nonsingular M-matrix and N?0. For an M-splitting of an irreducible Z-matrix A we prove that the circuit index of M-1N is the greatest common divisor of certain sets of integers associated with the circuits of A. For M-splittings of a reducible singular M-matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting. These results hold under somewhat weaker assumptions. |
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