15-01-ZeroMatrix.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ZeroMatrix}
\pmcreated{2013-03-22 14:19:19}
\pmmodified{2013-03-22 14:19:19}
\pmowner{waj}{4416}
\pmmodifier{waj}{4416}
\pmtitle{zero matrix}
\pmrecord{8}{35789}
\pmprivacy{1}
\pmauthor{waj}{4416}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-01}
\pmrelated{Matrix}
\pmrelated{IdentityMatrix}

\endmetadata

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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}

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 \newcommand{\Z}{\mathbbmss{Z}}
 \newcommand{\C}{\mathbbmss{C}}
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 \newcommand{\Q}{\mathbbmss{Q}}



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\begin{document}
\PMlinkescapeword{additive}
\PMlinkescapeword{identity}

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}
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\end{document}