Editing Zero matrix

{{Short description|Matrix whose entries are all 0}}
In [[mathematics]], particularly [[linear algebra]], a '''zero matrix''' or '''null matrix''' is a [[matrix (mathematics)|matrix]] all of whose entries are [[0 (number)|zero]]. It also serves as the [[additive identity]] of the [[additive group]] of <math>m \times n</math> matrices, and is denoted by the symbol <math>O</math> or <math>0</math> followed by subscripts corresponding to the dimension of the matrix as the context sees fit.<ref>{{citation|title=Linear Algebra|series=[[Undergraduate Texts in Mathematics]]|first=Serge|last=Lang|authorlink=Serge Lang|publisher=Springer|year=1987|isbn=9780387964126|page=25|url=https://books.google.com/books?id=0DUXym7QWfYC&pg=PA25|quotation=We have a zero matrix in which ''a<sub>ij</sub>''&nbsp;=&nbsp;0 for all ''i'',&nbsp;''j''. ... We shall write it&nbsp;''O''.}}</ref><ref>{{Cite web|title=Intro to zero matrices (article) {{!}} Matrices|url=https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:properties-of-matrix-addition-and-scalar-multiplication/a/intro-to-zero-matrices|access-date=2020-08-13|website=Khan Academy|language=en}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Zero Matrix|url=https://mathworld.wolfram.com/ZeroMatrix.html|access-date=2020-08-13|website=mathworld.wolfram.com|language=en}}</ref> Some examples of zero matrices are

:<math>
0_{1,1} = \begin{bmatrix}
0 \end{bmatrix}
,\ 
0_{2,2} = \begin{bmatrix}
0 & 0 \\
0 & 0 \end{bmatrix}
,\ 
0_{2,3} = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \end{bmatrix}
.\ 
</math>

==Properties==

The set of <math>m \times n</math> matrices with entries in a [[ring (mathematics)|ring]] K forms a ring <math>K_{m,n}</math>.  The zero matrix <math>0_{K_{m,n}} \, </math> in <math>K_{m,n} \, </math> is the matrix with all entries equal to <math>0_K \, </math>, where <math>0_K </math> is the [[additive identity]] in K.  

:<math>
0_{K_{m,n}} = \begin{bmatrix}
0_K & 0_K & \cdots & 0_K \\
0_K & 0_K & \cdots & 0_K \\
\vdots & \vdots & \ddots  & \vdots \\
0_K & 0_K & \cdots & 0_K \end{bmatrix}_{m \times n}
</math>

The zero matrix is the additive identity in <math>K_{m,n} \, </math>.<ref>{{citation|title=Modern Algebra|first=Seth|last=Warner|publisher=Courier Dover Publications|year=1990|isbn=9780486663418|page=291|url=https://books.google.com/books?id=dT2KAAAAQBAJ&pg=PA291|quotation=The neutral element for addition is called the zero matrix, for all of its entries are zero.}}</ref> That is, for all <math>A \in K_{m,n} \, </math> it satisfies the equation

:<math>0_{K_{m,n}}+A = A + 0_{K_{m,n}} = A.</math>

There is exactly one zero matrix of any given dimension ''m''&times;''n'' (with entries from a given ring), so when the context is clear, one often refers to ''the'' zero matrix.  In general, the [[zero element]] of a ring is unique, and is typically denoted by 0 without any [[subscript]] indicating the parent ring.  Hence the examples above represent zero matrices over any ring.

The zero matrix also represents the [[linear transformation]] which sends all the [[vector (geometric)|vector]]s to the [[zero vector]].<ref>{{citation|title=Linear Algebra: An Introduction|first1=Richard|last1=Bronson|first2=Gabriel B.|last2=Costa|publisher=Academic Press|year=2007|isbn=9780120887842|page=377|url=https://books.google.com/books?id=ZErjtA3mIvkC&pg=PA377|quotation=The zero matrix represents the zero transformation ''0'', having the property ''0''(''v'')&nbsp;=&nbsp;'''0''' for every vector ''v''&nbsp;∈&nbsp;''V''.}}</ref> It is [[idempotent matrix|idempotent]], meaning that when it is multiplied by itself, the result is itself.

The zero matrix is the only matrix whose [[rank (linear algebra)|rank]] is 0.

==Occurrences==

In [[ordinary least squares]] regression, if there is a perfect fit to the data, the [[annihilator matrix]] is the zero matrix.

==See also==
*[[Identity matrix]], the multiplicative identity for matrices
*[[Matrix of ones]], a matrix where all elements are one
*[[Nilpotent matrix]]
*[[Single-entry matrix]], a matrix where all but one element is zero

==References==
{{reflist}}

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:0 (number)]]
[[Category:Sparse matrices]]