Editing Block matrix (section)

{{Short description|Matrix defined using smaller matrices called blocks}}
In [[mathematics]], a '''block matrix''' or a '''partitioned matrix''' is a [[matrix (mathematics)|matrix]] that is interpreted as having been broken into sections called '''blocks''' or '''submatrices'''.<ref>{{cite book |last=Eves |first=Howard |author-link=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=[https://archive.org/details/elementarymatrix0000eves_r2m2/page/37 37] |url=https://archive.org/details/elementarymatrix0000eves_r2m2 |url-access=registration |edition=reprint |access-date=24 April 2013 |quote=We shall find that it is sometimes convenient to subdivide a matrix into rectangular blocks of elements. This leads us to consider so-called ''partitioned'', or ''block'', ''matrices''.}}</ref><ref name=":8">{{Cite web |last=Dobrushkin |first=Vladimir |date= |title=Partition Matrices |url=https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part2/partition.html |access-date=2024-03-24 |website=Linear Algebra with Mathematica}}</ref>

Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or [[Partition of a set|partition]] it, into a collection of smaller matrices.<ref>{{cite book |last=Anton |first=Howard |title=Elementary Linear Algebra |year=1994 |publisher=John Wiley |location=New York |isbn=0-471-58742-7 |page=30 |edition=7th |quote=A matrix can be subdivided or '''''partitioned''''' into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.}}</ref><ref name=":8" /> For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block.

:<math>
\left[
\begin{array}{ccc|c}
a_{11} & a_{12} & a_{13} & b_{1} \\
a_{21} & a_{22} & a_{23} & b_{2} \\
\hline
c_{1} & c_{2} & c_{3} & d
\end{array}
\right]
</math>

Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.

This notion can be made more precise for an <math>n</math> by <math>m</math> matrix <math>M</math> by partitioning <math>n</math> into a collection <math>\text{rowgroups}</math>, and then partitioning <math>m</math> into a collection <math>\text{colgroups}</math>. The original matrix is then considered as the "total" of these groups, in the sense that the <math>(i, j)</math> entry of the original matrix corresponds in a [[Bijection|1-to-1]] way with some <math>(s, t)</math> [[offset (computer science)|offset]] entry of some <math>(x,y)</math>, where <math>x \in \text{rowgroups}</math> and <math>y \in \text{colgroups}</math>.<ref>{{Cite journal |last1=Indhumathi |first1=D. |last2=Sarala |first2=S. |date=2014-05-16 |title=Fragment Analysis and Test Case Generation using F-Measure for Adaptive Random Testing and Partitioned Block based Adaptive Random Testing |url=http://research.ijcaonline.org/volume93/number6/pxc3895662.pdf |journal=International Journal of Computer Applications |volume=93 |issue=6 |pages=13 |doi=10.5120/16218-5662|bibcode=2014IJCA...93f..11I }}</ref>

Block matrix algebra arises in general from [[biproduct]]s in [[Category (mathematics)|categories]] of matrices.<ref>{{cite journal | last1 = Macedo | first1 = H.D. | last2 = Oliveira | first2 = J.N. | year = 2013 | title = Typing linear algebra: A biproduct-oriented approach | doi = 10.1016/j.scico.2012.07.012 | journal = Science of Computer Programming | volume = 78 | issue = 11| pages = 2160–2191 | arxiv = 1312.4818 }}</ref>
[[File:BlockMatrix168square.png|thumb|A 168×168 element block matrix with 12×12, 12×24, 24×12, and 24×24 sub-matrices. Non-zero elements are in blue, zero elements are grayed.]]