A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$.
Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$ is nilpotent
My try:
The Jordan Canonical form of the matrix of $T$ will consist of a Jordan block of order 2 corresponding to $\lambda =0$ and another Jordan block of order $1$.
Similarly The Jordan Canonical form of the matrix of $T$ will consist of a Jordan block of order $1$ corresponding to $\lambda =5$ and another Jordan block of order $1$.
How can I find from here the the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$ ?
Please give some hints.