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I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r \times n$ matrix.

Then some students raised this question: are there any applications of such factorization?

As I am not specialized in algebra, I have no knowledge about this. I have looked up some literatures, and found some corollaries of this factorization about idempotent matrics. But these sound unsatisfying for people who are non-mathematicians.

So, I am aksing whether there are "real" applications? e.g., solving some mathematical modelling problems, improving computational tasks.

P.S. this does not sound like a research question. But it seems improper to ask on stackexchange.

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closed as off-topic by Chris Godsil, Bill Johnson, Stefan Kohl, Olivier Benoist, Andrey Rekalo Jan 11 '14 at 12:47

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This decomposition can be used to give a short proof of Finsler's lemma, as can be seen in the appendix of (1) (Fortunately, the page that has the appendix is part of the preview offered by google books).

Finsler's lemma itself find many applications in control theory: several problems in control literature can be recast as a matrix inequality and Finsler's lemma can be used to add and remove variables from this matrix inequality. The books (2,3) and the references therein give a lot of examples of how to use this procedure.

It is worthy to remark that Finsler's lemma is also known as the Elimination Lemma when used in the context of removing variables from a matrix inequality.

(1) De Oliveira, M. C., & Skelton, R. E. (2001). Stability Tests for Constrained Linear Systems. In Perspectives in robust control (Vol. 268, pp. 241–257). London, UK: Springer.

(2) Boyd, S. P., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. SIAM. Available online at http://www.stanford.edu/~boyd/lmibook/.

(3) Skelton, R. E., Iwasaki, T., & Grigoriadis, K. (1997). A Unified Algebraic Approach to Control Design. London, UK: Taylor & Francis.

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  • $\begingroup$ Another interesting application of rank factorization is that it can be used to calculate the Cesàro limit of Markov chains: see pp.633-634 and pp.687-697 of Meyer, Carl D. Matrix analysis and applied linear algebra, SIAM (2000). $\endgroup$ – Shamisen Nov 19 '17 at 18:23
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There are many applications of this which have arisen as a result of the Netflix Prize. For a good set of references on such algorithms see A Comparative Study of Collaborative Filtering Algorithms.

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That factorization is almost the definition of being rank-$r$, so basically any result that involves a rank-$r$ matrix can be reformulated with little difficulty to use it, but it would be a bit artificial to say that "result X uses this factorization".

Moreover, it has too many degrees of freedom to be well-defined. A thin SVD, for instance, is a factorization in that format but uniquely defined.

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