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.[1]

J2=(0110)J3=(001010100)⋮Jn=(00⋯0100⋯10⋮⋮⋅⋅j˙⋮⋮01⋯0010⋯00)

## Contents

* 1 Definition
* 2 Properties
* 3 Relationships
* 4 See also
* 5 References

## Definition

If J is an n × n exchange matrix, then the elements of J are Ji,j={1,i+j=n+10,i+j≠n+1

## Properties

* Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,(001010100)(123456789)=(789456123).
* Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,(123456789)(001010100)=(321654987).
* Exchange matrices are symmetric; that is: JnT=Jn.
* For any integer k: Jnk={I if k is even,Jn if k is odd.In particular, Jn is an involutory matrix; that is, Jn−1=Jn.
* The trace of Jn is 1 if n is odd and 0 if n is even. In other words: tr⁡(Jn)=nmod2.
* The determinant of Jn is: det⁡(Jn)=(−1)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⁡(λI−Jn)={[(λ+1)(λ−1)]n2 if n is even,(λ−1)n+12(λ+1)n−12 if n is odd.
* The adjugate matrix of Jn is: adj⁡(Jn)=sgn⁡(πn)Jn. (where sgn is the sign of the permutation πk 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 = JAT 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), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885, https://books.google.com/books?id=O7sgAwAAQBAJ&pg=PA33 .

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Matrix classes

Explicitly constrained entries|

* (0,1)
* Alternant
* Anti-diagonal
* Anti-Hermitian
* Anti-symmetric
* Arrowhead
* Band
* Bidiagonal
* Binary
* 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
* Markov
* Metzler
* Monomial
* Moore
* Nonnegative
* Partitioned
* Parisi
* Pentadiagonal
* Permutation
* Persymmetric
* Polynomial
* Positive
* Quaternionic
* Sign
* Signature
* Skew-Hermitian
* Skew-symmetric
* Skyline
* Sparse
* Sylvester
* Symmetric
* Toeplitz
* Triangular
* Tridiagonal
* Unitary
* Vandermonde
* Walsh
* Z

Constant|

* Exchange
* Hilbert
* Identity
* Lehmer
* Of ones
* Pascal
* Pauli
* Redheffer
* Shift
* Zero

Conditions on eigenvalues or eigenvectors|

* Companion
* Convergent
* Defective
* Diagonalizable
* Hurwitz
* Positive-definite
* Stability
* Stieltjes

Satisfying conditions on products or inverses|

* Congruent
* Idempotent or Projection
* Invertible
* Involutory
* Nilpotent
* Normal
* Orthogonal
* Orthonormal
* Singular
* Unimodular
* Unipotent
* Totally unimodular
* Weighing

With specific applications|

* Adjugate
* Alternating sign
* Augmented
* Bézout
* Carleman
* Cartan
* Circulant
* Cofactor
* Commutation
* Confusion
* Coxeter
* Derogatory
* Distance
* Duplication
* Elimination
* Euclidean distance
* Fundamental (linear differential equation)
* Generator
* Gramian
* Hessian
* Householder
* Jacobian
* Moment
* Payoff
* Pick
* Random
* Rotation
* Seifert
* Shear
* Similarity
* Symplectic
* Totally positive
* Transformation
* Wedderburn
* X–Y–Z

Used in statistics|

* Bernoulli
* Centering
* Correlation
* Covariance
* Design
* Dispersion
* Doubly stochastic
* Fisher information
* Hat
* Precision
* Stochastic
* Transition

Used in graph theory|

* Adjacency
* Biadjacency
* Degree
* Edmonds
* Incidence
* Laplacian
* Seidel adjacency
* Skew-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 canonical form
* Linear independence
* Matrix exponential
* Matrix representation of conic sections
* Perfect matrix
* Pseudoinverse
* Quaternionic matrix
* Row echelon form
* Wronskian

* List of matrices
* Category:Matrices

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