Kunze Linear Algebra · Problem 6.4.2

The key insight is that any polynomial relation satisfied by $T$ on the whole space automatically restricts to a valid polynomial relation on $W$—since $W$ is invariant, $T$ maps $W$ into itself. If $m_T$ is the minimal polynomial of $T$, then $m_T(T)$ annihilates every vector in $V$, including all vectors in $W$. This means $m_T(T_W)$ annihilates $W$, so $m_T$ is a polynomial that kills $T_W$. The minimal polynomial $m_{T_W}$ is the smallest such polynomial, so it must divide $m_T$.

The key insight is that any polynomial relation satisfied by $T$ on the whole space automatically restricts to a valid polynomial relation on $W$—since $W$ is invariant, $T$ maps $W$ into itself. If $m_T$ is the minimal polynomial of $T$, then $m_T(T)$ annihilates every vector in $V$, including all vectors in $W$. This means $m_T(T_W)$ annihilates $W$, so $m_T$ is a polynomial that kills $T_W$. The minimal polynomial $m_{T_W}$ is the smallest such polynomial, so it must divide $m_T$.