# 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)