Content deleted Content added
Transpose |
m Added short description Tags: Mobile edit Mobile app edit iOS app edit App description add |
||
| (6 intermediate revisions by 5 users not shown) | |||
Line 1:
{{Short description|Formula for the pseudoinverse of a partitioned matrix}}
{{refimprove|date=December 2010}}
In [[mathematics]], a '''block matrix pseudoinverse''' is a formula for the [[pseudoinverse]] of a [[partitioned matrix]]. This is useful for decomposing or approximating many algorithms updating parameters in [[signal processing]], which are based on the [[least squares]] method.
Line 11 ⟶ 12:
</math>
If the above matrix is full column rank, the [[Moore–Penrose inverse]] matrices of it and its transpose are
:<math>\begin{align}
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^+ &=
Line 31 ⟶ 32:
This computation of the pseudoinverse requires (''n'' + ''p'')-square matrix inversion and does not take advantage of the block form.
To reduce computational costs to ''n''- and ''p''-square matrix inversions and to introduce parallelism, treating the blocks separately, one derives <ref name=Baksalary>{{cite journal|author=[[Jerzy Baksalary|J.K. Baksalary]] and O.M. Baksalary|title=Particular formulae for the Moore–Penrose inverse of a columnwise partitioned matrix|journal=Linear Algebra Appl.|volume=421|date=2007|pages=16–23|doi=10.1016/j.laa.2006.03.031|doi-access=}}</ref>
:<math>\begin{align}
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^+ &=
Line 177 ⟶ 178:
== See also ==
*
==References ==
Line 186 ⟶ 187:
* [https://web.archive.org/web/20060414125709/http://www.csit.fsu.edu/~burkardt/papers/linear_glossary.html Linear Algebra Glossary] by [http://www.csit.fsu.edu/~burkardt/ John Burkardt]
* [http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf The Matrix Cookbook] by [http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=3274/ Kaare Brandt Petersen]
* [http://see.stanford.edu/materials/lsoeldsee263/08-min-norm.pdf Lecture 8: Least-norm solutions of undetermined equations] by [
{{Numerical linear algebra}}
| |||