15-00-AntidiagonalMatrix.tex

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\pmtitle{anti-diagonal matrix}
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\pmtype{Definition}
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\begin{document}
Let $A$ be a square matrix (over any field $\F$). An entry in $A$ 
is an \emph{anti-diagonal entry} if it is on the line going from the 
lower left corner of $A$ to the upper right corner. If all 
entries in $A$ are zero except on the anti-diagonal, then $A$ is an 
\emph{anti-diagonal matrix}.

If $a_1, \ldots, a_n \in \F$, let 
$$
  \operatorname{adiag}(a_1, \ldots, a_n) = \begin{pmatrix}
0 & 0     & 0     & 0   & a_1 \\
0 & 0     & 0     & a_2 & 0  \\
0 & 0     & a_{3} &     & 0  \\
0 & \cdot & 0     &     & 0  \\
a_n & 0   & 0     &     &  0
\end{pmatrix}. 
$$

\subsubsection*{Properties of anti-diagonal matrices}
\begin{enumerate}
\item If $A$ and $D$ are $n\times n$ anti-diagonal and 
   diagonal matrices, respectively, then $AD, DA$ are anti-diagonal.
\item The product of two anti-diagonal matrices is an diagonal matrix.
\end{enumerate}
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