Perron transformation

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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 [1]. 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 [1], [2]. 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

is recurrent.


[1] O. Perron, "Ueber eine Matrixtransformation" Math. Z. , 32 (1930) pp. 465–473
[2] 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
[3] 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)
[4] 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
How to Cite This Entry:
Perron transformation. V.M. Millionshchikov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098