# Visualizing Operators on $\mathbb{C}^n$

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.

-

## 1 Answer

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 .

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9134198427200317, "perplexity_flag": "head"}