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]

[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]

## 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 [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, Jn is an involutory matrix; that is, [math]\displaystyle{ J_n^{-1} = J_n. }[/math]
* The trace of Jn 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 Jn 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 Jn 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 Jn is: [math]\displaystyle{ \operatorname{adj}(J_n) = \sgn(\pi_n) J_n. }[/math] (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 .

* v
* t
* e

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

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Exchange matrix. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Exchange_matrix&oldid=3388291"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template