Editing Singular value (section)

== The smallest singular value ==
The smallest singular value of a matrix ''A'' is ''σ''<sub>n</sub>(''A''). It has the following properties for a non-singular matrix A:

* The 2-norm of the inverse matrix A<sup>−1</sup> equals the inverse ''σ''<sub>n</sub><sup>−1</sup>(''A'').<ref name=":0">{{Cite book |last=Demmel |first=James W. |url=http://epubs.siam.org/doi/book/10.1137/1.9781611971446 |title=Applied Numerical Linear Algebra |date=January 1997 |publisher=Society for Industrial and Applied Mathematics |isbn=978-0-89871-389-3 |language=en |doi=10.1137/1.9781611971446}}</ref>{{Rp|location=Thm.3.3}}
* The absolute values of all elements in the inverse matrix A<sup>−1</sup> are at most the inverse ''σ''<sub>n</sub><sup>−1</sup>(''A'').<ref name=":0" />{{Rp|location=Thm.3.3}}

Intuitively, if ''σ''<sub>n</sub>(''A'') is small, then the rows of A are "almost" linearly dependent. If it is ''σ''<sub>n</sub>(''A'') = 0, then the rows of A are linearly dependent and A is not invertible.