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.

$${\displaystyle {\begin{aligned}J_{2}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\[4pt]J_{3}&={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}\\&\quad \vdots \\[2pt]J_{n}&={\begin{pmatrix}0&0&\cdots &0&1\\0&0&\cdots &1&0\\\vdots &\vdots &\,{}_{_{\displaystyle \cdot }}\!\,{}^{_{_{\displaystyle \cdot }}}\!{\dot {\phantom {j}}}&\vdots &\vdots \\0&1&\cdots &0&0\\1&0&\cdots &0&0\end{pmatrix}}\end{aligned}}}$$

If J is an n × n exchange matrix, then the elements of J are $${\displaystyle J_{i,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}}$$

its eigenvalues are 1 (with multiplicity ${\displaystyle \lceil n/2\rceil }$) and -1 (with multiplicity ${\displaystyle \lfloor n/2\rfloor }$).