Fundamental matrix and principal propagator

For continuous $A(t)$ , a fundamental matrix $X(t)$ is an invertible matrix whose columns solve the linear system $y'=A(t)y$ . Its associated principal propagator is

\Phi(t,s)=X(t)X(s)^{-1},\qquad \partial_t\Phi(t,s)=A(t)\Phi(t,s),\quad\Phi(s,s)=I.

It maps the state at time $s$ to the state at time $t$ , independently of the chosen fundamental matrix.

Composition and inverse

Uniqueness of the initial-value problem gives

\Phi(t,s)\Phi(s,r)=\Phi(t,r),\qquad \Phi(t,s)^{-1}=\Phi(s,t).

Differentiating the inverse yields $\partial_s\Phi(t,s)=-\Phi(t,s)A(s)$ . On a compact time interval, the Gronwall bound is $\|\Phi(t,s)\|\le\exp(\int_{\min(s,t)}^{\max(s,t)}\|A(\tau)\|\,d\tau)$ .