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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Functions of a real variable" , Addison-Wesley  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.P. Demidovich,   "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Yakubovich,   V.M. Starzhinskii,   "Linear differential equations with periodic coefficients" , Wiley  (1975)  (Translated from Russian)</TD></TR></table>

====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|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|Cauchy matrix]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.W. Brockett,   "Finite dimensional linear systems" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1980)</TD></TR></table>