Abstract:
The criterion of existence of nonlinear operator $\mathbf{u}:C^m\rightarrow C^m \left(m\ge2\right)$ for which it's Jacoby matrix commutes with every constant complex matrix from any given ring $Q$ is obtained. The main theorem says that such operator exists if and only if a ring $Q$ has at least one $\left(r,l\right)$-pair.
Keywords:criterion of existence of nonlinear operator, Jacoby matrix commuting with every constant complex matrix from any given ring, equivalence, $\boldsymbol{(r,l)}$-pair, Schur lemma.
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