Direct product/Eigenvectors/Exercise

Let ${}V_{1},\ldots ,V_{n}$ denote vector spaces over a field ${}K$, and let

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

denote linear mappings. Let ${}a\in K$ be an eigenvalue of ${}\varphi _{k}$ for a determined ${}k$. Show that ${}a$ is also an eigenvalue of the product mapping

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

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