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.