Editing Singular value decomposition (section)

===Thin SVD===
The thin, or economy-sized, SVD of a matrix {{tmath|\mathbf M}} is given by<ref>
{{cite book
| chapter-url = http://www.netlib.org/utk/people/JackDongarra/etemplates/node43.html
| contribution = Decompositions
| contributor-last = Demmel
| contributor-first = James
| last1 = Bai
| first1 = Zhaojun
| last2 = Demmel
| first2 = James
| last3 = Dongarra
| first3 = Jack J.
| last4 = Ruhe
| first4 = Axel
| last5 = van der Vorst
| first5 = Henk A.
| title = Templates for the Solution of Algebraic Eigenvalue Problems
| doi = 10.1137/1.9780898719581
| url = https://www.cs.ucdavis.edu/~bai/ET/contents.html
| year = 2000
| isbn = 978-0-89871-471-5
| publisher = Society for Industrial and Applied Mathematics
}}
</ref>

<math display=block>
\mathbf{M} = \mathbf{U}_k \mathbf \Sigma_k \mathbf{V}^*_k,
</math>

where <math>k = \min(m, n),</math> the matrices {{tmath|\mathbf U_k}} and {{tmath|\mathbf V_k}} contain only the first {{tmath|k}} columns of {{tmath|\mathbf U}} and {{tmath|\mathbf V,}} and {{tmath|\mathbf \Sigma_k}} contains only the first {{tmath|k}} singular values from {{tmath|\mathbf \Sigma.}} The matrix {{tmath|\mathbf U_k}} is thus {{tmath|m \times k,}} {{tmath|\mathbf \Sigma_k}} is {{tmath|k \times k}} diagonal, and {{tmath|\mathbf V_k^*}} is {{tmath|k \times n.}}

The thin SVD uses significantly less space and computation time if {{tmath|k \ll \max(m, n).}} The first stage in its calculation will usually be a [[QR decomposition]] of {{tmath|\mathbf M,}} which can make for a significantly quicker calculation in this case.