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I am trying to get some better intuition about operators on complex inner product spaces. When we identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, is there a nice geometric interpretation for the resulting operators on $\mathbb{R}^{2n}$? Ideally, this characterization would give a geometric construction for the adjoint generalizing conjugation as reflection in the real line. Also, "seeing" the polar decomposition would be nice.

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Unitary operators on $\mathbb{C}^n$ are realized as operators which are both orthogonal and symplectic on the correspoding $\mathbb{R}^{2n}$ (with a symplectic structure compatible to the original complex structure) . This is due to the 2-out-of-3 property of the unitary group.

Consequently, the polar decomposition expressed on $\mathbb{R}^{2n}$ has the structure:

$ A = U P$

where $U$ is an orthogonal and symplectic matrix on $\mathbb{R}^{2n}$ and $P$ is a positive $ 2n \times 2n$ matrix with multiplicity 2 eigenvalues. This can be seen easily through the realization of the polar decomposition through the singular value decomposition as given for example in the following Wikipedia page .

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