Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 41930, 69 pages
doi:10.1155/2007/41930

Fixed points of two-sided fractional matrix transformations

Abstract

Let C and D be n×n complex matrices, and consider the densely defined map φC,D:X↦(I−CXD)−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., CD−DC 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.