How to show that strictly hyperbolic implies strongly well posed?

Question

Consider a first order system $\partial_t u = P(D)u$ with $P$ given by $$ P(\xi) = \sum_{k=1}^n iA_k\xi_k. $$ Here, $n$ denotes the dimension of the spatial domain of $u$ (i.e. $\mathbb{R}^n$) and each $A_k \in {M}_{N\times N}(\mathbb{C})$ for some $N\in\mathbb{N}$. More precisely, consider solutions $u : \mathbb{R}^n \times \mathbb{R} \to \mathbb{C}^N$ to the system: $$ \partial_t u = P(D)u = \sum_{k=1}^n A_k \partial_ku. $$ I want to show that if $P$ is strictly hyperbolic (i.e. diagonalizable with purely imaginary and distinct eigenvalues) then the system above is strongly well posed. Now, I've reduced this to showing that, there exist constants $C, a$ such that $$ \left\lvert e^{tP(\xi)}\right \rvert \leq Ce^{a t} $$ for all $\xi\in\mathbb{R}^n$ and $t\geq 0$. In the above, $\left\lvert \cdot\right\rvert$ denotes the matrix norm. Does anyone have any advice on how to proceed?