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\title{Skew Transpose} % Declares the document's title.
\author{N. Setzer} % Declares the author's name.
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\maketitle
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\section{Definition and Notation}
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The Skew Transpose is the transpose of a square matrix about the other, non-main diagonal (top right to bottom left). If $A$ and $B$ are $n \times n$ matrices, the skew transpose is defined as
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$$
\inp{A^\$}_{ij} \equiv A_{(n + 1 - j)(n + 1 - i)}
$$
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\section{Properties}
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\subsection{$\inp{AB}^\$ = B^\$ A^\$$}
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\begin{eqnarray*}
\inb{\inp{AB}^\$}_{ij}
& = & \inp{AB}_{(n + 1 - j)(n + 1 - i)}
\\
& = & \sum_{k = 1}^n A_{(n + 1 - j)k} B_{k(n + 1 - i)}
\end{eqnarray*}
%
Now, $k$ is a dummy index that is summed over. As long as all the values it is summed over are hit, it can be expressed in any way that it convenient. The expression $n + 1 - k$ goes from $n$ to $1$ for $k$ going from $1$ to $n$, therefore, we can write
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\begin{eqnarray*}
\inb{\inp{AB}^\$}_{ij}
& = & \sum_{k = 1}^n A_{(n + 1 - j)(n + 1 - k)} B_{(n + 1 - k)(n + 1 - i)}
\\
& = & \sum_{k = 1}^n \inp{A^\$}_{kj} \inp{B^\$}_{ik}
\\
& = & \sum_{k = 1}^n \inp{B^\$}_{ik} \inp{A^\$}_{kj}
\\
& = & \inp{B^\$ A^\$}_{ij}
\end{eqnarray*}
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\subsection{Skew `Identity'}
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\newcommand{\SkewIdentity}{\mathbb{J}}
$$
\SkewIdentity =
\begin{pmatrix}
0 & 0 & \ldots & 0 & 1 \\
0 & 0 & \ldots & 1 & 0 \\
\vdots & \vdots& \adots & \vdots& \vdots\\
0 & 1 & \ldots & 0 & 0 \\
1 & 0 & \ldots & 0 & 0
\end{pmatrix}
$$
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$$
\SkewIdentity_{ij} = \delta_{n-i+1,j}
$$
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\Style{KroneckerDeltaUseComma=true,KroneckerDeltaIndicies=down}
$$
\begin{aligned}
\inp{\SkewIdentity^\$}_{ij}
& = \SkewIdentity_{n - j + 1, n - i + 1} \\
& = \KroneckerDelta{j, n - i + 1} \\
& = \KroneckerDelta{n - i + 1, j} \\
& = \SkewIdentity_{ij} \\
\end{aligned}
$$
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\subsection{$\inp{\Transpose{A}}^\$ = \Transpose{\inp{A^\$}} $}
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$$
\begin{aligned}
\inp{\Transpose{A}}^\$_{ij}
& = \inp{\Transpose{A}}_{n-j+1, n-i+1} \\
& = A_{n-i+1, n-j+1} \\
& = A^\$_{ji} \\
& = \Transpose{\inp{A^\$}}_{ij}
\end{aligned}
$$
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\subsection{$\SkewIdentity A \SkewIdentity = A^{\$ T}$}
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$$
\begin{aligned}
\inp{\SkewIdentity A \SkewIdentity}_{ij}
& = \KroneckerDelta{n-i+1,k} A_{km} \KroneckerDelta{n-m+1} \\
& = A_{n-i+1,m} \KroneckerDelta{m,n-j+1} \\
& = A_{n-i+1,n-j+1} \\
& = A^{\$ T}
\end{aligned}
$$
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