$B=C^{-1}AC=\left[ \begin{array}{cc} P & 0\\ 0 & Q \end{array} \right]$ consider the system y'=By+G(y).
Let $U(t)=\left[ \begin{array}{cc} e^{Pt} & 0\\ 0 & 0 \end{array} \right] $ and $V(t)=\left[ \begin{array}{cc} 0 & 0\\ 0 & e^{Qt} \end{array} \right] $ where $P$ is $k\times k$ has eigenvalues with negative real part and $Q$ is $(n-k) \times (n-k)$ has eigenvalues with positive real part. For $\alpha>0$ sufficiently small that for $j=1,..,k$ we choose $Re(\lambda_{j})<-\alpha<0$. We have $e^{Bt}=U(t)+V(t)$
Why in the proof it was written that for $K>0$ and $\delta <0$ we have $$ ||U(t)||\le Ke^{-(\alpha+\delta)t} \text{ for all $t\ge 0$} $$ and $$ ||V(t)||\le Ke^{\delta t} \text{ for all $t\le 0$} $$