Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set $W ( t ) = \operatorname { det } ( E ( t ) )$ and denote $\rho _ { 1 }$ and $\rho _ { 2 }$ the Floquet multipliers of (IV.1). Deduce that $\rho _ { 1 } \rho _ { 2 } = \exp \left( \int _ { 0 } ^ { T } \operatorname { tr } ( A ( s ) ) \mathrm { d } s \right)$.
Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system
$$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$
We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set $W ( t ) = \operatorname { det } ( E ( t ) )$ and denote $\rho _ { 1 }$ and $\rho _ { 2 }$ the Floquet multipliers of (IV.1). Deduce that $\rho _ { 1 } \rho _ { 2 } = \exp \left( \int _ { 0 } ^ { T } \operatorname { tr } ( A ( s ) ) \mathrm { d } s \right)$.