zero matrix

The $n\times m$ zero $O$ over a ring $R$ is the $n\times m$ matrix withcoefficients in $R$ given by

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:

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

• $O$ has only one eigenvalue $\lambda =0$ ofmultiplicity $n$. Any non-zero vector is an eigenvector of $O$, so if we’re looking for a basis of eigenvectors, we could pick the standard basis $e_1=(1,0,\mathrm{\dots },0),\mathrm{\dots },e_n=(0,\mathrm{\dots },0,1)$.

• The matrix exponential of $O$ is $e^O=I$, the $n\times n$ identity matrix.