Abstract
AbstractLet "Equation missing" and "Equation missing" be "Equation missing" complex matrices, and consider the densely defined map "Equation missing" on "Equation missing" matrices. Its fixed points form a graph, which is generically (in terms of "Equation missing") nonempty, and is generically the Johnson graph "Equation missing"; 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 "Equation missing" and "Equation missing" are entrywise nonnegative and "Equation missing" 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., "Equation missing" is in the radical of the algebra generated by "Equation missing" and "Equation missing") are discussed in detail. For invertible size two matrices, a fixed point exists for all choices of "Equation missing" if and only if "Equation missing" 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.
Highlights
Let C and D be square complex matrices of size n
We obtain a densely defined mapping from the set of n × n matrices to itself, φC,D : X → (I − CXD)−1. We refer to this as a two-sided matrix fractional linear transformation, these really only correspond to the denominator of the standard fractional linear transformations, z →
(az + b)/(cz + d) (apparently more general transformations, such as X → (CXD + E)−1, reduce to the ones we study here). These arise in the determination of harmonic functions of fairly natural infinite state Markov chains [1]
Summary
Let C and D be square complex matrices of size n. We obtain a densely defined mapping from the set of n × n matrices (denoted MnC) to itself, φC,D : X → (I − CXD)−1. If there is exactly one, φC,D has no attractive fixed points at all, and the unique positive one has a “flow-through” property (inspired by a type of tea bag). This leads to a numerical invariant for nonnegative matrices, which, is difficult to calculate (except when the matrix is normal). The sets of fixed points of these (compositions) can be transformed to their counterparts for φC,D
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