Difference between revisions of "Perron transformation"
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An orthogonal (unitary) 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 | + | 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 | 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 [[#References|[1]]]. Perron's theorem applies: For any linear system (2) with continuous coefficients | + | It was introduced by O. Perron [[#References|[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|orthogonalization]] (for each | + | A Perron transformation is constructed by means of Gram–Schmidt [[Orthogonalization|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|fundamental system of solutions]] to (2), where different fundamental systems give, in general, different Perron transformations [[#References|[1]]], [[#References|[2]]]. For systems (2) with bounded continuous coefficients, all the Perron transformations are Lyapunov transformations (cf. [[Lyapunov transformation|Lyapunov transformation]]). | ||
| − | If the matrix-valued function | + | If the matrix-valued function $ \| a _ {j} ^ {i} ( t) \| $, |
| + | $ i, j = 1 \dots n $, | ||
| + | is a [[Recurrent function|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. | is recurrent. | ||
Latest revision as of 19:31, 11 January 2024
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 |
How to Cite This Entry:
Perron transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron_transformation&oldid=15417
Perron transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron_transformation&oldid=15417
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article