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]

$${\displaystyle {\begin{aligned}J_{2}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\[4pt]J_{3}&={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}\\&\quad \vdots \\[2pt]J_{n}&={\begin{pmatrix}0&0&\cdots &0&1\\0&0&\cdots &1&0\\\vdots &\vdots &\,{}_{_{\displaystyle \cdot }}\!\,{}^{_{_{\displaystyle \cdot }}}\!{\dot {\phantom {j}}}&\vdots &\vdots \\0&1&\cdots &0&0\\1&0&\cdots &0&0\end{pmatrix}}\end{aligned}}}$$

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: $${\displaystyle J_{n}^{\mathsf {T}}=J_{n}.}$$
• For any integer k: $${\displaystyle J_{n}^{k}={\begin{cases}I&{\text{ if }}k{\text{ is even,}}\\[2pt]J_{n}&{\text{ if }}k{\text{ is odd.}}\end{cases}}}$$In particular, J_n is an involutory matrix; that is, $${\displaystyle J_{n}^{-1}=J_{n}.}$$
• The trace of J_n is 1 if n is odd and 0 if n is even. In other words: $${\displaystyle \operatorname {tr} (J_{n})={\frac {1-(-1)^{n}}{2}}=n{\bmod {2}}.}$$
• The determinant of J_n is: $${\displaystyle \det(J_{n})=(-1)^{\lfloor n/2\rfloor }=(-1)^{\frac {n(n-1)}{2}}}$$ 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: $${\displaystyle \det(\lambda I-J_{n})=(\lambda -1)^{\lceil n/2\rceil }(\lambda +1)^{\lfloor n/2\rfloor }={\begin{cases}{\big [}(\lambda +1)(\lambda -1){\big ]}^{\frac {n}{2}}&{\text{ if }}n{\text{ is even,}}\\[4pt](\lambda -1)^{\frac {n+1}{2}}(\lambda +1)^{\frac {n-1}{2}}&{\text{ if }}n{\text{ is odd,}}\end{cases}}}$$

its eigenvalues are 1 (with multiplicity ${\displaystyle \lceil n/2\rceil }$) and -1 (with multiplicity ${\displaystyle \lfloor n/2\rfloor }$).

• The adjugate matrix of J_n is: $${\displaystyle \operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}.}$$ (where sgn is the sign of the permutation π_k of k elements).

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.

See also

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

References

1. Horn, Roger A.; Johnson, Charles R. (2012), "§0.9.5.1 n-by-n reversal matrix", Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN   978-1-139-78888-5 .