Editing Matrix unit

{{Short description|Concept in mathematics}}
{{Distinguish|unit matrix|unitary matrix|invertible matrix}}
In [[linear algebra]], a '''matrix unit''' is a [[matrix (mathematics)|matrix]] with only one nonzero entry with value 1.<ref>{{cite book |last=Artin|first=Michael |title=Algebra |publisher= Prentice Hall|page=9}}</ref><ref name="pm">{{cite book |chapter=Chapter 17: Matrix Rings |title=Lectures on Modules and Rings |first=Tsit-Yuen |last=Lam |authorlink=Tsit-Yuen Lam |series=[[Graduate Texts in Mathematics]] |volume=189 |publisher=[[Springer Science+Business Media]] |year=1999 |pages=461–479}}</ref>  The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as <math>E_{ij}</math>. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is
<math display=block>E_{12} = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math>A '''vector unit''' is a [[standard unit vector]].

A '''single-entry matrix''' generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.

== Properties ==

The set of ''m'' by ''n'' matrix units is a [[basis (linear algebra)|basis]] of the space of ''m'' by ''n'' matrices.<ref name="pm"/>

The product of two matrix units of the same square shape <math>n \times n</math> satisfies the relation
<math display=block>E_{ij}E_{kl} = \delta_{jk}E_{il},</math>
where <math>\delta_{jk}</math> is the [[Kronecker delta]].<ref name="pm"/>

The group of [[scalar matrix|scalar]] ''n''-by-''n'' matrices over a ring ''R'' is the [[centralizer]] of the subset of ''n''-by-''n'' matrix units in the set of ''n''-by-''n'' matrices over ''R''.<ref name="pm"/>

The [[matrix norm]] (induced by the same two vector norms) of a matrix unit is equal to 1.

When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix ''A'':<ref>{{Cite arXiv
 | author = Marcel Blattner
 | title = B-Rank: A top N Recommendation Algorithm
 | year = 2009
 | class = physics.data-an
 | eprint = 0908.2741
 }}</ref>
: <math>
 E_{23}A = \left[ \begin{matrix} 0 & 0& 0 \\ a_{31} & a_{32} & a_{33} \\ 0 & 0 & 0 \end{matrix}\right].
</math>
: <math>
 AE_{23} = \left[ \begin{matrix} 0 & 0 & a_{12} \\ 0 & 0 & a_{22} \\ 0 & 0 & a_{32}  \end{matrix}\right].
</math>

==References==
{{reflist}}

{{matrix classes}}

[[Category:Sparse matrices]]
[[Category:1 (number)]]

{{Linear-algebra-stub}}