Exchange matrix

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.

$$\begin{align} J_2 &= \begin{pmatrix} 0 & 1 \\   1 & 0  \end{pmatrix} \\[4pt] J_3 &= \begin{pmatrix} 0 & 0 & 1 \\   0 & 1 & 0 \\    1 & 0 & 0  \end{pmatrix} \\ &\quad \vdots \\[2pt] J_n &= \begin{pmatrix} 0     & 0      & \cdots      & 0      & 1      \\ 0     & 0      & \cdots      & 1      & 0      \\ \vdots & \vdots & \,{}_{_{\displaystyle\cdot}} \!\, {}^{_{_{\displaystyle\cdot}}} \! \dot\phantom{j} & \vdots & \vdots \\ 0     & 1      & \cdots      & 0      & 0      \\ 1     & 0      & \cdots      & 0      & 0 \end{pmatrix} \end{align}$$

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

Properties
\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} = \begin{pmatrix} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{pmatrix}. $$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 3 & 2 & 1 \\ 6 & 5 & 4 \\ 9 & 8 & 7 \end{pmatrix}. $$ J_n^\mathsf{T} = J_n.$$ J_n^k = \begin{cases} I & \text{ if } k \text{ is even,} \\[2pt] J_n & \text{ if } k \text{ is odd.} \end{cases} $$In particular, $k$ is an involutory matrix; that is, $$ J_n^{-1} = J_n.$$ \operatorname{tr}(J_n) = n\bmod 2.$$ \det(J_n) = (-1)^\frac{n(n-1)}{2} $$ As a function of $J_{n}$, it has period 4, giving 1, 1, −1, −1 when $J_{n}$ is congruent modulo 4 to 0, 1, 2, and 3 respectively. \det(\lambda I- J_n) = \begin{cases} \big[(\lambda+1)(\lambda-1)\big]^\frac{n}{2} & \text{ if } n \text{ is even,} \\[4pt] (\lambda-1)^\frac{n+1}{2}(\lambda+1)^\frac{n-1}{2} & \text{ if } n \text{ is odd.} \end{cases}$$ \operatorname{adj}(J_n) = \sgn(\pi_n) J_n. $$ (where $sgn$ is the sign of the permutation $n$ of $n$ elements).
 * Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,$$
 * Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,$$
 * Exchange matrices are symmetric; that is: $$
 * For any integer $J_{n}$: $$
 * The trace of $n$ is 1 if $n$ is odd and 0 if $J_{n}$ is even. In other words: $$
 * The determinant of $J_{n}$ is: $$
 * The characteristic polynomial of $&pi;k$ is: $$
 * The adjugate matrix of $k$ is: $$

Relationships

 * An exchange matrix is the simplest anti-diagonal matrix.
 * Any matrix $A$ satisfying the condition $AJ = JA$ is said to be centrosymmetric.
 * Any matrix $A$ satisfying the condition $AJ = JA^{T}$ is said to be persymmetric.
 * Symmetric matrices $A$ that satisfy the condition $AJ = JA$ are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.