Abstract:
We expose the complete integration of the simplest matrix Riccati equation in the two- and three-dimensional cases for an arbitrary linear differential operator. The solution is constructed in terms of the Jordan form of an unknown matrix and the corresponding similarity matrix. We show that a similarity matrix is always representable as the product of two matrices one of which is an invariant of the differential operator.
Keywords:
matrix Riccati equation, algebraic invariant, Jordan form.
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The authors were supported by the Programs of Basic Research nos. III.22.4.1 and
I.1.5 (project no. 0314-2019-0011) of the Siberian Branch of the Russian Academy of
Sciences.
Citation:
M. V. Neshchadim, A. P. Chupakhin, “On integration of a matrix Riccati equation”, Sib. Zh. Ind. Mat., 23:4 (2020), 101–113; J. Appl. Industr. Math., 14:4 (2020), 732–742
This publication is cited in the following 5 articles:
Yu. E. Anikonov, M. V. Neschadim, A. P. Chupakhin, “Mnogomernoe uravnenie Khopfa i nekotorye ego tochnye resheniya”, Sib. zhurn. industr. matem., 25:1 (2022), 5–13
S. A. Vasyutkin, A. P. Chupakhin, “Postroenie minimalnogo bazisa invariantov dlya differentsialnoi algebry (2×2)-matrits”, Sib. zhurn. industr. matem., 25:2 (2022), 21–31
S. A. Vasyutkin, A. P. Chupakhin, “Constructing a Minimal Basis of Invariants for Differential Algebra of 2×2 Matrices”, J. Appl. Ind. Math., 16:2 (2022), 356
Yu. E. Anikonov, M. V. Neshchadim, A. P. Chupakhin, “Multidimensional Hopf Equation and Some of Its Exact Solutions”, J. Appl. Ind. Math., 16:1 (2022), 1
M. V. Neshchadim, A. P. Chupakhin, “Method of commutators for integration of a matrix Riccati equation”, J. Appl. Industr. Math., 15:1 (2021), 78–86