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If $S$ and $T$ are commuting, normal operators, then $ST$ is normal

That says it all, but let me be more specific. (By the way Wikipedia says this: "The product of normal operators that commute is again normal; this is nontrivial and follows from Fuglede's theorem")

Let $V$ be a finite dimensional inner-product space. Do not assume that it is complex. Suppose that $S$ and $T$ are normal operators that commute. Prove that $ST$ is normal.

This result is relatively simple if we had assumed that our inner-product space was complex. Just by using the spectral theorem, we can simultaneously diagonalize $S$ and $T$ and we would be done.

The question is, then, how does one show this for a real inner-product space. One suggestion is to complexify our space --- but I am not too familiar with this. Does anyone care to give this a shot?

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Has you already solved your question? – rafaeldf Feb 25 '14 at 19:11

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