Fundamental matrix
matrizant
The transition matrix of solutions of a system of linear ordinary differential equations
![]() | (*) |
normalized at the point . The fundamental matrix is the unique continuous solution of the matrix initial value problem
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( denotes the identity matrix) if the matrix-valued function
is locally summable over some interval
,
.
Every matrix built of column-solutions
of the system (*), where
is a natural number, is expressible as
. In particular, every solution
of (*) can be written in the form
.
The expansion
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which converges absolutely for every and uniformly on every compact interval in
, and the Liouville–Ostrogradski formula
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are valid. If the matrix satisfies the Lappo-Danilevskii condition
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then
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In particular, if is a constant matrix, then
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If is the fundamental matrix of the system (*) with matrix
, then
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where
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The fundamental matrix makes it possible to write every solution of the inhomogeneous system
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in which the function is locally summable on
, in the form of Cauchy's formula
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here
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is called the Cauchy matrix of (*). The Cauchy matrix is jointly continuous in its arguments on
and for arbitrary
it has the properties
1) ;
2) ;
3) ;
4) ;
5) ,
, where
is the norm in
;
6) if is the Cauchy matrix of the adjoint system
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then
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References
[1] | N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French) |
[2] | F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) |
[3] | B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian) |
[4] | V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients" , Wiley (1975) (Translated from Russian) |
Comments
The term "matrizant" is no longer in common use; instead the term "transition matrixtransition matrix" has become popular for what is called above "fundamental matrix" . See also Fundamental system of solutions.
Cauchy's formula is often called the variation of constants formula, and the Cauchy matrix is also called the transition matrix (cf. also Cauchy matrix).
References
[a1] | R.W. Brockett, "Finite dimensional linear systems" , Wiley (1970) |
[a2] | J.K. Hale, "Ordinary differential equations" , Wiley (1980) |
Fundamental matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_matrix&oldid=14038