Perron transformation

An orthogonal (unitary) transformation

$$ \tag{1 } x ^ {i} = \sum_{j=1}^ { n } u _ {j} ^ {i} ( t) y ^ {j} ,\ \ i = 1 \dots n, $$

smoothly depending on $ t $ and transforming a linear system of ordinary differential equations

$$ \tag{2 } \dot{x} ^ {i} = \sum_{j=1} ^ { n } a _ {j} ^ {i} ( t) x ^ {j} ,\ \ i = 1 \dots n, $$

to a system of triangular type

$$ \tag{3 } \dot{y} ^ {i} = \sum_{j=i} ^ { n } p _ {j} ^ {i} ( t) y ^ {j} ,\ \ i = 1 \dots n. $$

It was introduced by O. Perron [1]. Perron's theorem applies: For any linear system (2) with continuous coefficients $ a _ {j} ^ {i} ( t) $, a Perron transformation exists.

A Perron transformation is constructed by means of Gram–Schmidt orthogonalization (for each $ t $) of the vector system $ x _ {1} ( t) \dots x _ {n} ( t) $, where $ x _ {1} ( t) \dots x _ {n} ( t) $ 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 $ \| a _ {j} ^ {i} ( t) \| $, $ i, j = 1 \dots n $, is a recurrent function, one can find a recurrent matrix-valued function $ \| u _ {j} ^ {i} ( t) \| $, $ i, j = 1 \dots n $, such that (1) is the Perron transformation that reduces (2) to the triangular form (3), where, moreover, the function

$$ \| p _ {j} ^ {i} ( t) \| ,\ \ i, j = 1 \dots n, $$

is recurrent.

References

[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