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]

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

Definition

If J is an n×n exchange matrix, then the elements of J are defined such that:

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

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 ${\displaystyle (-1)^{n(n-1)/2}}$. As a function of n, it has period 4, giving 1, 1, −1, −1 when ${\displaystyle n{\bmod {4}}=0,1,2,3}$, 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.

See also

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

References

1. Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885.