Exponential matrix definition

Question

The exponential matrix is $e^{tA} : = X(t)$ where $X$ is the unique global solution of the differential equation $x'=Ax$ which satisfies $X(0)=I$. I want to prove, using this definition, that $$ e^{(st)A} = e^{s(tA)}.$$ Any ideas?

Answer 1

Let $X_1$ the solution of $X'=AX$ with $X(0)=I$.

Let $s \in \mathbb{R}$ and $X_s$ the solution of $X'=sAX$ with $X(0)=I$

Then by definition $X_1(t) = e^{tA}$ and $X_s(t) = e^{t(sA)}$

The goal is to show $X_1(st)=X_s(t)$ for all $t$.

Let $Y_s$ defined by $Y_s(t)=X_1(st)$.

We have $Y_s(0)= I$ and $Y_s'(t)=sX_1'(st)=sAX_1(st)=sAY_s(t)$.
Hence $Y$ is solution of $X'=sAX$ with $X(0)=I$.

By unicity of the solution, $Y_s=X_s$, that is $X_1(st)=X_s(t)$.

Hence $e^{stA}= e^{t(sA)} \space \forall s,t$ Q.E.D