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A criterion for the existence of a nonlinear mapping whose Jacobian matrix commutes with a matrix ring

Abstract

Let Q be a ring of constant square matrices of orderm over the field of complex numbers. We consider the problem on the existence of a nonlinear mapping u: C mC m, m ≥ 2, whose Jacobian matrix commutes with each matrix of Q. We prove that such a mapping exists if and only if Q possesses an (r, l)-pair.

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References

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Correspondence to Yu. A. Chirkunov.

Additional information

Original Russian Text © Yu. A. Chirkunov, 2010, published in Vestnik NGU. Matematika, Mekhanika, Informatika, 2010, Vol. 10, No. 1, pp. 108–118.

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Chirkunov, Y.A. A criterion for the existence of a nonlinear mapping whose Jacobian matrix commutes with a matrix ring. Sib. Adv. Math. 21, 250–258 (2011). https://doi.org/10.3103/S105513441104002X

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Keywords

  • nonlinear mapping
  • Jacobian matrix
  • matrix ring
  • commutation condition
  • compatibility condition
  • (r, l)-pair
  • composition series
  • Schur’s lemma