I've learned the Primary decomposition theorem recently and wanted to check if I have this right. Is this example correctly applied?
Let $V$ be finite dim. and $T$ a linear map on $V$.
Suppose $m_T(x)=x(x-1)^5(x^2+1)^2$ is the minimal polynomial of $T$.
Then by the PDT, $V=\ker T \bigoplus \ker (T-I)^5 \bigoplus \ker (T^2+I)^2$
and $m_{T|_{\ker (T-I)}}=(x-1)^5$.