Exchange matrix

Short description: Square matrix whose entries are 1 along the anti-diagonal and 0 elsewhere

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.

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Definition

If J is an n × n exchange matrix, then the elements of J are [math]\displaystyle{ J_{i,j} = \begin{cases} 1, & i + j = n + 1 \\ 0, & i + j \ne n + 1\\ \end{cases} }[/math]

Properties

• Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,[math]\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}. }[/math]
• Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,[math]\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}. }[/math]
• Exchange matrices are symmetric; that is: [math]\displaystyle{ J_n^\mathsf{T} = J_n. }[/math]
• For any integer k: [math]\displaystyle{ J_n^k = \begin{cases} I & \text{ if } k \text{ is even,} \\[2pt] J_n & \text{ if } k \text{ is odd.} \end{cases} }[/math]In particular, J_n is an involutory matrix; that is, [math]\displaystyle{ J_n^{-1} = J_n. }[/math]
• The trace of J_n is 1 if n is odd and 0 if n is even. In other words: [math]\displaystyle{ \operatorname{tr}(J_n) = n\bmod 2. }[/math]
• The determinant of J_n is: [math]\displaystyle{ \det(J_n) = (-1)^\frac{n(n-1)}{2} }[/math] 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: [math]\displaystyle{ \det(\lambda I- J_n) = \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} }[/math]
• The adjugate matrix of J_n is: [math]\displaystyle{ \operatorname{adj}(J_n) = \sgn(\pi_n) J_n. }[/math] (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)