# Orthonormality proof on outer product of SVD unitary matrices columns

If I have the two unitary matrices from the SVD of an m x n matrix (U, V*) and I form a set of new matrices by doing $u_iv_i^H$ (forms an m x n matrix). Assuming $r = min(m, n)$ and my set is $X_1, X_2, ..., X_r$, how can I show that this is an orthonormal set? Is there some property of unitary matrices that I've forgotten? Their rows and columns form an orthonormal basis in $C^n$ but what of the product of two orthonormal vectors?

Furthermore, how do I generalize the coordinates of the original matrix using this orthonormal basis of the vector space?

-
What are $X_k$, the dyads $u_kv_k^*$? If so, what do you mean by orthonormal (for matrices)? – copper.hat Dec 13 '12 at 19:29
By orthonormal matrices, I mean the set is orthogonal and the matrix norm (generalization of the vector norm) is 1. – PatternMatching Dec 13 '12 at 19:43

1
I understand the definition of orthonormal. I was asking what inner product you are using for two matrices? One definition is $\langle A, B \rangle = \mathbb{tr} A^*B$, and with this inner product the $X_k$ are orthonormal. – copper.hat Dec 13 '12 at 20:08
Sorry - yes, the inner product would be as you have specified above. – PatternMatching Dec 13 '12 at 20:17

## 1 Answer

If you let $X_{ij} = u_i v_j^*$, and define the inner product $\langle A, B \rangle = \mathbb{tr} A^*B$, then you have $\langle X_{ij}, X_{ab} \rangle = \delta_{ia} \delta_{jb}$, from which it follows that the $X_{ij}$ form an orthonormal basis (there are $mn$ of them).

The collection of matrices you have above is a subset of this collection, so they are orthonormal (but not a basis).

-
{"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": 13, "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": 0.9069189429283142, "perplexity_flag": "head"}