Exchange matrix

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title: "Exchange matrix" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["matrices-(mathematics)"] description: "Square matrix whose entries are 1 along the anti-diagonal and 0 elsewhere" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Exchange_matrix" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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.

\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}

Definition

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

Properties

• Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e., \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., \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: J_n^\mathsf{T} = J_n.
• For any integer k: 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, Jn is an involutory matrix; that is, J_n^{-1} = J_n.
• The trace of Jn is 1 if n is odd and 0 if n is even. In other words: \operatorname{tr}(J_n) = \frac{1-(-1)^n}{2} = n\bmod 2.
• The determinant of Jn is: \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 Jn is: \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 \lceil n/2\rceil) and -1 (with multiplicity \lfloor n/2\rfloor).
• The adjugate matrix of Jn is: \operatorname{adj}(J_n) = \sgn(\pi_n) J_n. (where sgn is the sign of the permutation π of k elements).

Relationships

• An exchange matrix is the simplest anti-diagonal matrix.
• Any matrix A satisfying the condition is said to be centrosymmetric.
• Any matrix A satisfying the condition is said to be persymmetric.
• Symmetric matrices A that satisfy the condition are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.

References

References

1. (2012). "Matrix Analysis". Cambridge University Press.

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