Exchange matrix

In mathematics, especially linear algebra, the exchange matrix (also called the reversal matrix, backward identity, or standard involutory permutation) is a special case of a permutation matrix, where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix.[1]

J_{2}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};\quad J_{3}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}};\quad J_{n}={\begin{pmatrix}0&0&\cdots &0&0&1\\0&0&\cdots &0&1&0\\0&0&\cdots &1&0&0\\\vdots &\vdots &&\vdots &\vdots &\vdots \\0&1&\cdots &0&0&0\\1&0&\cdots &0&0&0\end{pmatrix}}.

Definition

If J is an n×n exchange matrix, then the elements of J are defined such that:

J_{i,j}={\begin{cases}1,&j=n-i+1\\0,&j\neq n-i+1\\\end{cases}}

Properties

J^T = J.
Jⁿ = I for even n; Jⁿ = J for odd n, where n is any integer. In particular, J is an involutory matrix; that is, J⁻¹ = J.
The trace of J is 1 if n is odd, and 0 if n is even.
The characteristic polynomial of J is $\det(\lambda I-J_{n})={\big (}(1-\lambda )(1+\lambda ){\big )}^{n/2}$ for $n$ even, and $(1-\lambda )^{(n+1)/2}(1+\lambda )^{(n-1)/2}$ for $n$ odd.
The adjugate matrix of J is $\operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}$ .

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.

References

Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885.

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[1] Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885.

Exchange matrix

Definition

Properties

Relationships

See also

References