Geodesic
On integration of a matrix Riccati equation
Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 4, pp. 101-113.

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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.
Mots-clés : matrix Riccati equation, algebraic invariant, Jordan form. . 
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M. V. Neshchadim; A. P. Chupakhin. On integration of a matrix Riccati equation. Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 4, pp. 101-113. http://geodesic.mathdoc.fr/item/SJIM_2020_23_4_a7/

[1] M. I. Zelikin, Homogeneous spaces and the riccati equation in the calculus of variations, Faktorial, M., 1998 (in Russian)

[2] F. A. Chernous'ko, V. B. Kolmanovskii, Optimal control under random perturbations, Nauka, M., 1978 (in Russian) | MR

[3] L. V. Ovsyannikov, Lectures on the fundamentals of gas dynamics, Inst. Kompyuternykh Issledovanii, M.–Izhevsk, 2003 (in Russian) | MR

[4] A. P. Chupakhin, Barochronous gas motions: general properties and submodels of types (1, 2) and (1, 1), Preprint No 4-98, Lavrent'ev Inst. Gidrodin. SO RAN, Novosibirsk, 1998 (in Russian)

[5] A. P. Chupakhin, Preprint No 1-99, Lavrent'ev Inst. Gidrodin. SO RAN, Novosibirsk, 1999 (in Russian)

[6] A. A. Cherevko, A. P. Chupakhin, The Stationary Ovsyannikov vortex, Preprint No 1-2005, Lavrent-ev Inst. Gidrodin. SO RAN, Novosibirsk, 2005 (in Russian) | MR

[7] E. Miller, “A Regularity Criterion for the Navier-Stokes equation involving only the middle eigenvalue of the strain tensor”, Arch. Rational Mech. Anal., 235 (2020), 99–139 | DOI | MR | Zbl

[8] C. A. Stephen, M. W. Gary, “Hierarchies of new invariants and conserved integrals in inviscid fluid flow”, Phys. Fluids, 32 (2020), 086104 | DOI | MR

[9] A. G. Fat'yanov, “Some semianalytical method for solution of the direct dynamical problems in stratified media”, Dokl. Akad. Nauk SSSR, 310:2 (1990), 323–327 (in Russian) | MR

[10] A. L. Kapchevckii, “Analiticheckoe peshenie upavnenii Makcvella v chactotnoi oblacti dlya gopizontal-nocloictyx anizotpopnyx cped [Analytical solution of the Maxwell equations in the frequency domain for horizontal layered anisotropic media]”, Geologiya i geofizika, 48:8 (2007), 889–898

[11] A. L. Karchevsky, B. R. Rysbayuly, “Analitical expressions for a solution of convective heat and moisture transfer equations in the frequency domain for layered media”, Euras. J. Math. Comp. Appl., 3:4 (2015), 55–67

[12] A. L. Karchevskii, “Analytical solutions of the differential equation of transverse vibrations of a piecewise homogeneous beam in the frequency domain for the boundary conditions of various types”, J. Appl. Ind. Math., 14:4 (2020), 650–668 | MR

[13] J. Peyrière, “On an Article by W. Magnus on the Fricke Characters of Free Groups”, J. Algebra, 228 (2020), 659–673 | DOI | MR

[14] W. Magnus, “Rings of Fricke characters and automorphism groups of free groups”, Math. Zh., 170 (1980), 91–103 | DOI | MR | Zbl

[15] C. Procesi, “The invariant theory of $n\times n$ matrices”, Adv. Math., 19 (1976), 306–381 | DOI | MR | Zbl

[16] Yu. P. Razsmyslov, “Trace identities of complete matrix algebras over a field of zero characteristic”, Izv. AN SSSR. Ser. Matem., 38:4 (1974), 723–756 (in Russian)

[17] A. Whittemore, “On special linear characters of free groups of rank $n \geqslant 4 $”, Proc. Amer. Math. Soc., 40 (1973), 383–388 | MR | Zbl

[18] Y. Avishai, D. Berend, V. Tkachenko, “Trace maps”, Internat. J. Modern Phys. B, 11 (1997), 3525–3542 | DOI | MR | Zbl

[19] M. Bresara, C. Procesi, S. Spenko, “Quasi-identities on matrices and the Cayley-Hamilton polynomial”, Adv. Math., 280 (2015), 439–471 | DOI | MR

[20] M. D. Cvetković, L. S. Velimirović, “Application of shape operator under infinitesimal bending of surface filomat”, Filomat, 33:4 (2019), 1267–1271 | DOI | MR

[21] L. S. Velimirović, M. D. Cvetković, N. M. Velimirovic, M. S. Najdanović, “Variation of shape operator under infinitesimal bending of surface”, Appl. Math. Comput., 225 (2013), 480–486 | MR | Zbl

[22] K. K. Brustad, Total derivatives of eigenvalues and eigenprojections of symmetric matrices, 15 May 2019, arXiv: 1905.06045v1 [math.AP]

[23] N. J. Rose, “On the eigenvalues of a matrix which commutes with its derivative”, Proc. Amer. Math. Soc., 4 (1965), 752–754 | DOI | MR

[24] I. J. Epstein, “Conditions for a matrix to commute with its integral”, Proc. Amer. Math. Soc., 14 (1963), 266–270 | DOI | MR | Zbl

[25] M. Hausner, “Eigenvalues of certain operators on matrices”, Comm. Pure Appl. Math., 14 (1961), 155–156 | DOI | MR | Zbl

[26] F. R. Gantmakher, AMS Chelsea Publ, Providence, 1998 | MR | MR