zero matrixZeroMatrix2004-04-21 12:57:392004-04-25 12:28:43Definition% this is the default PlanetMath preamble. as your knowledge
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The $n \times m$ \emph{zero \PMlinkescapetext{matrix}} $O$ over a ring $R$ is the $n \times m$ matrix with
coefficients in $R$ given by
$$ O =
\begin{bmatrix}
0 & \cdots & 0 \\
\vdots & \ddots & \vdots \\
0 & \cdots & 0 \\
\end{bmatrix},$$
where 0 is the \PMlinkname{additive identity}{Ring} in $R$.
\subsubsection{Properties}
The zero matrix is the additive identity in the ring of $n\times n$ matrices over $R$. This \PMlinkescapetext{property} is an alternative definition of $O$ (since there's \PMlinkname{just one additive identity in any given ring}{UniquenessOfAdditiveIdentityInARing2}).
The $n\times n$ zero matrix $O$ has the following properties:
\begin{itemize}
\item The determinant of $O$ is $\det O = 0$, and its trace is
$\operatorname{tr}O = 0$.
\item $O$ has only one eigenvalue $\lambda =0$ of
multiplicity $n$. 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 $e_1=(1,0,\ldots, 0), \ldots , e_n=(0,\ldots, 0,1)$.
\item The matrix exponential of $O$ is $e^O = I$, the $n\times n$ identity matrix.
\end{itemize}