Difference between revisions of "Perron transformation"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | p0723901.png | ||
+ | $#A+1 = 13 n = 0 | ||
+ | $#C+1 = 13 : ~/encyclopedia/old_files/data/P072/P.0702390 Perron transformation | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
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. |
Revision as of 08:05, 6 June 2020
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 |
Perron transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron_transformation&oldid=48167