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Reference.org
Exchange matrix
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of <a href="/facts/Permutation_matrix/xC1ran5s">permutation matrices</a>, where the 1 elements reside on the <a href="/facts/Main_diagonal/rkT2F08Q">antidiagonal</a> and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. </p><p> J 2 = ( 0 1 1 0 ) J 3 = ( 0 0 1 0 1 0 1 0 0 ) ⋮ J n = ( 0 0 ⋯ 0 1 0 0 ⋯ 1 0 ⋮ ⋮ ⋅ ⋅ j ˙ ⋮ ⋮ 0 1 ⋯ 0 0 1 0 ⋯ 0 0 ) {\displaystyle {\begin{aligned}J_{2}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\[4pt]J_{3}&={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}\\&\quad \vdots \\[2pt]J_{n}&={\begin{pmatrix}0&0&\cdots &0&1\\0&0&\cdots &1&0\\\vdots &\vdots &\,{}_{_{\displaystyle \cdot }}\!\,{}^{_{_{\displaystyle \cdot }}}\!{\dot {\phantom {j}}}&\vdots &\vdots \\0&1&\cdots &0&0\\1&0&\cdots &0&0\end{pmatrix}}\end{aligned}}} </p>