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Difference between revisions of "Perron transformation"

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$$ \tag{1 }
 
$$ \tag{1 }
x  ^ {i}  =  \sum _ { j= } 1 ^ { n }  u _ {j}  ^ {i} ( t) y  ^ {j} ,\ \  
+
x  ^ {i}  =  \sum_{j=1}^ { n }  u _ {j}  ^ {i} ( t) y  ^ {j} ,\ \  
 
i = 1 \dots n,
 
i = 1 \dots n,
 
$$
 
$$
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$$ \tag{2 }
 
$$ \tag{2 }
\dot{x}  ^ {i}  =  \sum _ { j= } 1 ^ { n }  a _ {j}  ^ {i} ( t) x  ^ {j} ,\ \  
+
\dot{x}  ^ {i}  =  \sum_{j=1} ^ { n }  a _ {j}  ^ {i} ( t) x  ^ {j} ,\ \  
 
i = 1 \dots n,
 
i = 1 \dots n,
 
$$
 
$$
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$$ \tag{3 }
 
$$ \tag{3 }
\dot{y}  ^ {i}  =  \sum _ { j= } i ^ { n }  p _ {j}  ^ {i} ( t) y  ^ {j} ,\ \  
+
\dot{y}  ^ {i}  =  \sum_{j=i} ^ { n }  p _ {j}  ^ {i} ( t) y  ^ {j} ,\ \  
 
i = 1 \dots n.
 
i = 1 \dots n.
 
$$
 
$$

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=54991
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article