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 m → C 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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Yu. A. Chirkunov, “Linear autonomy conditions for the basic Lie algebra of a system of linear differential equations,” Dokl. Akad. Nauk 426(5), 605–607 (2009) [Dokl.Math. 79 (3), 415–417 (2009)].
Yu. A. Chirkunov, “Systems of linear differential equations symmetric with respect to transformations nonlinear in a function,” Sibirsk. Mat. Zh. 50(3), 680–686 (2009) [SiberianMath. J. 50 (3), 541–546 (2009)].
H. Weyl, Group Theory and Quantum Mechanics (Dutton, New York, 1931).
D. P. Zhelobenko and A. I. Shtern, Representations of Lie Groups (Nauka, Moscow, 1983) [in Russian].
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
- nonlinear mapping
- Jacobian matrix
- matrix ring
- commutation condition
- compatibility condition
- (r, l)-pair
- composition series
- Schur’s lemma