Editing Zero matrix (section)

==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.