Square matrix whose entries are 1 along the anti-diagonal and 0 elsewhere
In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the identity matrix.[1]
![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c783ae3381dc05095dbcc9b73eea1dd3d4b7bd0)
Definition
If J is an n × n exchange matrix, then the elements of J are
![{\displaystyle J_{i,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffed71fdfa68189c7d6b2eb73dc8bbdd2eb5b768)