An orthogonal (unitary) transformation
smoothly depending on and transforming a linear system of ordinary differential equations
to a system of triangular type
It was introduced by O. Perron . Perron's theorem applies: For any linear system (2) with continuous coefficients , a Perron transformation exists.
A Perron transformation is constructed by means of Gram–Schmidt orthogonalization (for each ) of the vector system , where is some fundamental system of solutions to (2), where different fundamental systems give, in general, different Perron transformations , . For systems (2) with bounded continuous coefficients, all the Perron transformations are Lyapunov transformations (cf. Lyapunov transformation).
If the matrix-valued function , , is a recurrent function, one can find a recurrent matrix-valued function , , such that (1) is the Perron transformation that reduces (2) to the triangular form (3), where, moreover, the function
|||O. Perron, "Ueber eine Matrixtransformation" Math. Z. , 32 (1930) pp. 465–473|
|||S.P. Diliberto, "On systems of ordinary differential equations" S. Lefschetz (ed.) et al. (ed.) , Contributions to the theory of nonlinear oscillations , Ann. Math. Studies , 20 , Princeton Univ. Press (1950) pp. 1–38|
|||B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)|
|||N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1976) pp. 45–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146|
Perron transformation. V.M. Millionshchikov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Perron_transformation&oldid=15417