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.[1]

\( {\displaystyle J_{2}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};\quad J_{3}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}};\quad J_{n}={\begin{pmatrix}0&0&\cdots &0&0&1\\0&0&\cdots &0&1&0\\0&0&\cdots &1&0&0\\\vdots &\vdots &&\vdots &\vdots &\vdots \\0&1&\cdots &0&0&0\\1&0&\cdots &0&0&0\end{pmatrix}}.} \)

Definition

If J is an n × n exchange matrix, then the elements of J are

\({\displaystyle J_{i,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}} \)

Properties

Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,

\( {\displaystyle {\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}={\begin{pmatrix}7&8&9\\4&5&6\\1&2&3\end{pmatrix}}.} \)

Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,

\( {\displaystyle {\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}3&2&1\\6&5&4\\9&8&7\end{pmatrix}}.} \)

Exchange matrices are symmetric; that is, JnT = Jn.
For any integer k, J_n^k = I if k is even and J_n^k = J_n if k is odd. In particular, J_n is an involutory matrix; that is, J_n⁻¹ = J_n..
The trace of J_n is 1 if n is odd and 0 if n is even. In other words, the trace of J_n equals \( {\displaystyle n{\bmod {2}}} \).
The determinant of J_n equals \( {\displaystyle (-1)^{n(n-1)/2}} \).As a function of n, it has period 4, giving 1, 1, −1, −1 when n is congruent modulo 4 to 0, 1, 2, and 3 respectively.
The characteristic polynomial of J_n is \( {\displaystyle \det(\lambda I-J_{n})={\big (}(\lambda +1)(\lambda -1){\big )}^{n/2}} \) when n is even, and \( {\displaystyle (\lambda -1)^{(n+1)/2}(\lambda +1)^{(n-1)/2}} \) when n is odd.
The adjugate matrix of J_n is \( {\displaystyle \operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}} \).

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.