Endomorphism/Orthogonal sum/Normal/Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let

${}V=V_{1}\oplus V_{2}\,$

denote a direct sum decomposition into linear subspaces ${}V_{1}$ and ${}V_{2}$, which are orthogonal to each other. Let

$\varphi _{1}\colon V_{1}\longrightarrow V_{1}$

and

$\varphi _{2}\colon V_{2}\longrightarrow V_{2}$

be normal endomorphisms, and

${}\varphi =\varphi _{1}\oplus \varphi _{2}\,$

their direct sum. Show that also ${}\varphi$ is normal.

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