What's wrong in my example about Q-matrix? I find the next property about Q-matrices:
$$ \text{Let Q a Q-Matrix, then}
\frac{d^k P(0)}{dt^k}=Q^k
 ; \text{ for }  k=0,1,2,\cdots.$$ I was trying to verify the property with the next example:
$Q=\begin{pmatrix}
-1 &amp; 1 \\
1 &amp; -1
\end{pmatrix}$ then
$$P(s)=\begin{pmatrix}
e^{-s} &amp; e^s \\
e^s &amp; e^{-s}
\end{pmatrix}
$$
$$\frac{d^k P(0)}{ds^k}=\begin{pmatrix}
(-1)^ke^{-0} &amp; e^0 \\
e^0 &amp; (-1)^ke^{-0}
\end{pmatrix}=\begin{pmatrix}
(-1)^k &amp; 1 \\
1 &amp; (-1)^k
\end{pmatrix}\neq Q^k$$
But it contradicts the proposition!! I want to know what is wrong in my example or interpretation.
If $P(s)$ and $Q$ are as above, then we have

$\frac{d^k P(0)}{ds^k}=Q^k$ only for $k=1,3,5,...$.
If $P(s)$ and $Q$ are as above, then we have

$\frac{d^k P(0)}{ds^k}=Q^k$ only for $k=1,3,5,...$.