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.

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

Definition

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

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

Properties

• Exchange matrices are symmetric; that is, J_n^T = J_n.
• For any integer k, J_n^k = I 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. (Same to say the trace equals to $n\%2$.)
• 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 is congruent modulo 4 to 0, 1, 2, and 3 respectively.
• The characteristic polynomial of J_n is $\det(\lambda I-J_{n})={\big (}(\lambda +1)(\lambda -1){\big )}^{n/2}$ when n is even, and $(\lambda -1)^{(n+1)/2}(\lambda +1)^{(n-1)/2}$ when n is odd.
• The adjugate matrix of J_n is $\operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}$.

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.

See also

• Pauli matrices (the first Pauli matrix is a 2 × 2 exchange matrix)