Fixed Points of Two-Sided Fractional Matrix Transformations
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Bibliographic record
Abstract
Abstract Let "Equation missing"<!-- image only, no MathML or LaTex --> and "Equation missing"<!-- image only, no MathML or LaTex --> be "Equation missing"<!-- image only, no MathML or LaTex --> complex matrices, and consider the densely defined map "Equation missing"<!-- image only, no MathML or LaTex --> on "Equation missing"<!-- image only, no MathML or LaTex --> matrices. Its fixed points form a graph, which is generically (in terms of "Equation missing"<!-- image only, no MathML or LaTex -->) nonempty, and is generically the Johnson graph "Equation missing"<!-- image only, no MathML or LaTex -->; 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"<!-- image only, no MathML or LaTex --> and "Equation missing"<!-- image only, no MathML or LaTex --> are entrywise nonnegative and "Equation missing"<!-- image only, no MathML or LaTex --> 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"<!-- image only, no MathML or LaTex --> is in the radical of the algebra generated by "Equation missing"<!-- image only, no MathML or LaTex --> and "Equation missing"<!-- image only, no MathML or LaTex -->) are discussed in detail. For invertible size two matrices, a fixed point exists for all choices of "Equation missing"<!-- image only, no MathML or LaTex --> if and only if "Equation missing"<!-- image only, no MathML or LaTex --> 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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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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