"Let $ A \in \mathbb{R^{n \times n}} $. If $ 0 \in \mathbb{R^{n}} $ is an attractor of $A$, show that, for a large enough $ \tau > 0$, the norm $ \Vert x \Vert$ in $\mathbb{R^{n}}$ defined by: $$\Vert x \Vert = \int_{0}^{ \tau } |e^{sA}x|ds$$ satisfies, for some $ \lambda >0 $: $$ \Vert e^{tA}x \Vert \leq \Vert x \Vert e^{- \lambda t}$$"
I have two basic ideas:
I know that $0$ being an attractor of $A$ is equivalent to stating that $\exists \mu > 0, k \geq 1 \text{ such that } |e^{tA}x| \leq ke^{- \mu t} |x|, \forall t \in \mathbb{R}, \forall x \in \mathbb{R^{n}} $. This result wasn't proved for this particular norm, but considering norm equivalence in euclidean spaces then I think the inequality should translate accordingly. Right now the problem is the factor $k$ in the inequality, since the result would just be a direct consequence if I could prove that $k=1$. Now, from the "for a large enough $ \tau > 0$" statement, I got the following idea: Obviously $|e^{sA}x| \geq 0$ for any value of $s$. Also, since $0$ can't be an eigenvalue of $e^{sA}$, when $x$ isn't the zero vector, then $|e^{sA}x| > 0$. That means: $$\int_{0}^{ \tau } |e^{sA}x|ds < \int_{0}^{ \tau + \alpha} |e^{sA}x|ds, \forall \alpha > 0$$ Let's call $$\Vert x \Vert_{ \tau} = \int_{0}^{ \tau } |e^{sA}x|ds$$ Then: $$\Vert x \Vert_{ \tau} < \Vert x \Vert_{ \tau + \alpha}, \forall \alpha >0$$ Thus, from $\Vert e^{tA}x \Vert_{ \tau} \leq k \Vert x \Vert_{ \tau} e^{ - \mu t}$ (with $k > 1$) I think I should find an $ \alpha \in \mathbb{R^{+}}$ such that $k \Vert x \Vert_{ \tau} = \Vert x \Vert_{ \tau + \alpha}$. I don't know how to find this relation between $k$ and $ \alpha $ (or if I'm approaching the problem from the correct direction), though.
I know the matrix operator norm is submultiplicative, as a consequence of its definition $ \Vert A \Vert_{op} = sup \{ \frac{\Vert Av \Vert}{\Vert v \Vert}, v \neq 0\}$. Then, $\Vert x \Vert \Vert e^{tA} \Vert_{op} \geq \Vert e^{tA}x \Vert$. However, I don't know how to estimate the value of this operator norm, in order to preserve the inequality.
Any hints would be appreciated.