Endomorphism/Direct sum/Minimal polynomial/Ideal intersection/Exercise

Let ${}V$ and ${}W$ be finite-dimensional ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow V$

and

$\psi \colon W\longrightarrow W$

be endomorphisms, with the minimal polynomials ${}P$ and ${}Q$. Show that the minimal polynomial of

$\varphi \oplus \psi \colon V\oplus W\longrightarrow V\oplus W$

equals the normed generator of the ideal ${}(P)\cap (Q)$.

Create a solution