Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 41930, 69 pages
Research Article

Fixed Points of Two-Sided Fractional Matrix Transformations

David Handelman

Mathematics Department, University of Ottawa, Ottawa K1N 6N5, ON, Canada

Received 16 March 2006; Revised 19 November 2006; Accepted 20 November 2006

Academic Editor: Thomas Bartsch

Copyright © 2007 David Handelman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let C and D be n×n complex matrices, and consider the densely defined map φC,D:X(ICXD)1 on n×n matrices. Its fixed points form a graph, which is generically (in terms of (C,D)) nonempty, and is generically the Johnson graph J(n,2n); in the nongeneric case, either it is a retract of the Johnson graph, or there is a topological continuum of fixed points. Criteria for the presence of attractive or repulsive fixed points are obtained. If C and D are entrywise nonnegative and CD is irreducible, then there are at most two nonnegative fixed points; if there are two, one is attractive, the other has a limited version of repulsiveness; if there is only one, this fixed point has a flow-through property. This leads to a numerical invariant for nonnegative matrices. Commuting pairs of these maps are classified by representations of a naturally appearing (discrete) group. Special cases (e.g., CDDC is in the radical of the algebra generated by C and D) are discussed in detail. For invertible size two matrices, a fixed point exists for all choices of C if and only if D has distinct eigenvalues, but this fails for larger sizes. Many of the problems derived from the determination of harmonic functions on a class of Markov chains.