Direct sum/Diagonalizable/Exercise

Let ${}V_{1},\ldots ,V_{n}$ be finite-dimensional vector spaces over the field ${}K$, and let

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

be linear mappings. Let

$\varphi =\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}$

be the product mapping. Show that ${}\varphi$ is diagonalizable if and only if all ${}\varphi _{i}$ are diagonalizable.

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