# 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) |

**How to Cite This Entry:**

Fundamental matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fundamental_matrix&oldid=47026