Endomorphism/Direct sum/Minimal polynomial and characteristic polynomial/Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Let

$\psi =\varphi \oplus \cdots \oplus \varphi \colon V\oplus \cdots \oplus V\longrightarrow V\oplus \cdots \oplus V$

be the ${}m$-fold direct sum of ${}\varphi$ with itself. How is the minimal polynomial (the characteristic polynomial) of ${}\psi$ related to the minimal polynomial (the characteristic polynomial) of ${}\varphi$?

Create a solution