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.



Applications of rank factorization or full rank decomposition
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](http://books.google.com.br/books?id=mz2r1XJqtPAC&pg=PA256 "this page")).


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.


