Exchange matrix

In mathematics, especially linear algebra, the exchange matrix (also called the reversal matrix, backward identity, or standard involutory permutation) is a special case of a permutation matrix, where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix.^[1]

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 defined such that:

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

Properties

Exchange matrices are symmetric; that is, J_n^T = J_n.
For any integer k, J_n^k = I for even k; J_n^k = J_n for odd k. 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.
The determinant of J_n equals $(-1)^{n(n-1)/2}$ . As a function of n, it has period 4, giving 1, 1, -1, -1 when $n{\bmod {4}}=0,1,2,3$ , respectively.
The characteristic polynomial of J_n is $\det(\lambda I-J_{n})={\big (}(1-\lambda )(1+\lambda ){\big )}^{n/2}$ when n is even, and $(1-\lambda )^{(n+1)/2}(1+\lambda )^{(n-1)/2}$ when n is odd.
The adjugate matrix of J_n is $\operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}$ .