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| {{Unreferenced|date=November 2006}}
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| In [[mathematics]], especially [[linear algebra]], the '''exchange matrix''' is a special case of a [[permutation matrix]], where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the [[identity matrix]].
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| :<math>
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| J_{2}=\begin{pmatrix}
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| 0 & 1 \\
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| 1 & 0
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| \end{pmatrix};\quad J_{3}=\begin{pmatrix}
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| 0 & 0 & 1 \\
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| 0 & 1 & 0 \\
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| 1 & 0 & 0
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| \end{pmatrix};
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| \quad J_{n}=\begin{pmatrix}
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| 0 & 0 & \cdots & 0 & 0 & 1 \\
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| 0 & 0 & \cdots & 0 & 1 & 0 \\
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| 0 & 0 & \cdots & 1 & 0 & 0 \\
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| \vdots & \vdots & & \vdots & \vdots & \vdots \\
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| 0 & 1 & \cdots & 0 & 0 & 0 \\
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| 1 & 0 & \cdots & 0 & 0 & 0
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| \end{pmatrix}.
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| </math>
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| ==Definition==
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| If ''J'' is an ''n×n'' exchange matrix, then the elements of ''J'' are defined such that:
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| :<math>J_{i,j} = \begin{cases}
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| 1, & j = n - i + 1 \\
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| 0, & j \ne n - i + 1\\
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| \end{cases}</math>
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| ==Properties==
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| * ''J''<sup>T</sup> = ''J''.
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| * ''J<sup>n</sup>'' = ''I'' for even ''n''; ''J<sup>n</sup>'' = ''J'' for odd ''n'', where ''n'' is any integer. Thus ''J'' is an [[involutary matrix]]; that is, ''J''<sup>−1</sup> = ''J''.
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| * The [[trace (linear algebra)|trace]] of ''J'' is ''1'' if ''n'' is [[Even and odd numbers|odd]], and ''0'' if ''n'' is [[Even and odd numbers|even]].
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| ==Relationships==
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| * Any matrix ''A'' satisfying the condition ''AJ = JA'' is said to be [[centrosymmetric matrix|centrosymmetric]].
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| * Any matrix ''A'' satisfying the condition ''AJ = JA''<sup>T</sup> is said to be [[persymmetric matrix|persymmetric]].
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| {{DEFAULTSORT:Exchange Matrix}}
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| [[Category:Matrices]]
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| {{Linear-algebra-stub}}
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