Fixed Points of Two-Sided Fractional Matrix Transformations
© David Handelman. 2007
Received: 16 March 2006
Accepted: 20 November 2006
Published: 28 February 2007
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.
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