Exchange matrix

Last updated January 02, 2025

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]

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

Definition

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

Properties

Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e., ${\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}={\begin{pmatrix}7&8&9\\4&5&6\\1&2&3\end{pmatrix}}.$
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, $J n$ 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. In other words: $\operatorname {tr} (J_{n})={\frac {1-(-1)^{n}}{2}}=n{\bmod {2}}.$
The determinant of $J n$ 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 $J n$ 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 $J n$ is: $\operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}.$ (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.

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References

↑ 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 .

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[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 .

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v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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