Suppose that $T:V \longrightarrow V$ and $T':W \longrightarrow W$ be linear operators on $V$ and $W$ respectively. Let $V'=V \oplus W$ and let $T'':V' \longrightarrow V'$ be a linear opearator on $V'$. If $m_T (t)$ and $m_{T'}(t)$ be respectively denote the minimal polynomials of $T$ and $T'$ respectively. Then what will be the minimal polynomial for $T''$?
I found that if $m_{T''}(t)$ denotes the minimal polynomial of $T''$ then $m_{T''} (t) = \mathrm {lcm}\ \{m_T (t) , m_{T'} (t) \}$. But can we go further? By "further" I mean to say that can I say that $\mathrm {lcm}\ \{m_T (t) , m_{T'} (t) \} = m_T (t) m_{T'} (t)$? If so why? Please help me.
Thank you in advance.
No, the new minimum polynomial is the lcm of the original ones, but need not be the product. For instance, if $W$ is a copy of $V$ and $T'$ is the same as $T$, then $m_{T'}=m_T$ and $\text{lcm}(m_T,m_{T'})=m_T\ne m_Tm_{T'}$.