Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 41930, 69 pages
Fixed Points of Two-Sided Fractional Matrix Transformations
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 and be complex matrices, and
consider the densely defined map on matrices. Its fixed points form a graph, which is generically
(in terms of ) nonempty, and is generically the Johnson graph
; 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 and are entrywise nonnegative and 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., is in the radical of the algebra generated
by and ) are discussed in detail. For invertible size two matrices,
a fixed point exists for all choices of if and only if 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.