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

( 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

which converges absolutely for every and uniformly on every compact interval in , and the Liouville–Ostrogradski formula

are valid. If the matrix satisfies the Lappo-Danilevskii condition

then

In particular, if is a constant matrix, then

If is the fundamental matrix of the system (*) with matrix , then

where

The fundamental matrix makes it possible to write every solution of the inhomogeneous system

in which the function is locally summable on , in the form of Cauchy's formula

here

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

then

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)