Fundamental matrix

matrizant

The transition matrix $ X( t) $ of solutions of a system of linear ordinary differential equations

$$ \tag{* } \dot{x} = A( t) x,\ \ x \in \mathbf R ^ {n} , $$

normalized at the point $ t _ {0} $. The fundamental matrix is the unique continuous solution of the matrix initial value problem

$$ \dot{X} = A( t) X,\ \ X( t _ {0} ) = I $$

( $ I $ denotes the identity matrix) if the matrix-valued function $ A( t) $ is locally summable over some interval $ J \subset \mathbf R $, $ t \in J $.

Every matrix $ M( t) $ built of column-solutions $ x _ {1} \dots x _ {m} $ of the system (*), where $ m $ is a natural number, is expressible as $ M( t) = X( t) M( t _ {0} ) $. In particular, every solution $ x( t) $ of (*) can be written in the form $ x( t) = X( t) x _ {0} $.

The expansion

$$ X( t) = I + \int\limits _ {t _ {0} } ^ { t } A( s) ds + \int\limits _ {t _ {0} } ^ { t } A( s) \int\limits _ {t _ {0} } ^ { s } A( r) dr ds + \dots , $$

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

$$ \mathop{\rm det} X( t) = \mathop{\rm exp} \int\limits _ {t _ {0} } ^ { t } \mathop{\rm Sp} A( s) ds $$

are valid. If the matrix $ A( t) $ satisfies the Lappo-Danilevskii condition

$$ A( t) \cdot \int\limits _ {t _ {0} } ^ { t } A( s) ds = \ \int\limits _ {t _ {0} } ^ { t } A( s) ds \cdot A( t), $$

then

$$ X( t) = \mathop{\rm exp} \int\limits _ {t _ {0} } ^ { t } A( s) ds. $$

In particular, if $ A( t) \equiv A $ is a constant matrix, then

$$ X( t) = e ^ {A( t- t _ {0} ) } . $$

If $ X _ {A} ( t) $ is the fundamental matrix of the system (*) with matrix $ A( t) $, then

$$ X _ {A+} B ( t) = X _ {A} ( t) X _ {D} ( t), $$

where

$$ D( t) = [ X _ {A} ( t)] ^ {-} 1 B( t) X _ {A} ( t). $$

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

$$ \dot{x} = A( t) x + b( t) , $$

in which the function $ b( t) $ is locally summable on $ J $, in the form of Cauchy's formula

$$ x( t) = X( t) x( t _ {0} ) + \int\limits _ {t _ {0} } ^ { t } C( t, s) b( s) ds,\ \ t \in J; $$

here

$$ C( t, s) = X( t)[ X( s)] ^ {-} 1 $$

is called the Cauchy matrix of (*). The Cauchy matrix $ C( t, s) $ is jointly continuous in its arguments on $ J \times J $ and for arbitrary $ t, s, r \in J $ it has the properties

1) $ C( t, s) = C( t, t _ {0} ) [ C( s, t _ {0} ) ] ^ {-} 1 $;

2) $ C( t, s) = C( t, r) C( r, s) $;

3) $ C( s, t) = [ C( t, s)] ^ {-} 1 $;

4) $ C( t, t) = I $;

5) $ | C( t, s) | \leq \mathop{\rm exp} \int _ {s} ^ {t} | A( r) | dr $, $ s \leq t $, where $ | \cdot | $ is the norm in $ \mathbf R ^ {n} $;

6) if $ H( t, s) $ is the Cauchy matrix of the adjoint system

$$ \dot{x} = - A ^ {*} ( t) x, $$

then

$$ H( t, s) = [ C ^ {*} ( t, s)] ^ {-} 1 . $$

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)