zero matrix
The $n\times m$ zero $O$ over a ring $R$ is the $n\times m$ matrix with coefficients in $R$ given by
$$O=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \end{array}\right],$$ 
where 0 is the additive identity (http://planetmath.org/Ring) in $R$.
0.0.1 Properties
The zero matrix^{} is the additive identity in the ring of $n\times n$ matrices over $R$. This is an alternative definition of $O$ (since there’s just one additive identity in any given ring (http://planetmath.org/UniquenessOfAdditiveIdentityInARing2)).
The $n\times n$ zero matrix $O$ has the following properties:

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The determinant^{} of $O$ is $detO=0$, and its trace is $\mathrm{tr}O=0$.

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$O$ has only one eigenvalue^{} $\lambda =0$ of multiplicity $n$. Any nonzero vector is an eigenvector^{} of $O$, so if we’re looking for a basis of eigenvectors, we could pick the standard basis ${e}_{1}=(1,0,\mathrm{\dots},0),\mathrm{\dots},{e}_{n}=(0,\mathrm{\dots},0,1)$.

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The matrix exponential^{} of $O$ is ${e}^{O}=I$, the $n\times n$ identity matrix^{}.
Title  zero matrix 

Canonical name  ZeroMatrix 
Date of creation  20130322 14:19:19 
Last modified on  20130322 14:19:19 
Owner  waj (4416) 
Last modified by  waj (4416) 
Numerical id  8 
Author  waj (4416) 
Entry type  Definition 
Classification  msc 1501 
Related topic  Matrix 
Related topic  IdentityMatrix 