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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]
Definition[edit]
If J is an n × n exchange matrix, then the elements of J are
Properties[edit]
- Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,
![{\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}}.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/1399bb3b1fa8edd6818cd48c891a895904eecdd4)
- Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,
![{\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}}.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ffccd2590153e42411787069dbb5ea14946c5117)
- Exchange matrices are symmetric; that is:
![{\displaystyle J_{n}^{\mathsf {T}}=J_{n}.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/624c9612ba58b440932ca71c46efc7d8c648a065)
- For any integer k:
In particular, Jn is an involutory matrix; that is, ![{\displaystyle J_{n}^{-1}=J_{n}.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/4489576587141503fc13aab32bb47f501c18a4f6)
- The trace of Jn is 1 if n is odd and 0 if n is even. In other words:
![{\displaystyle \operatorname {tr} (J_{n})=n{\bmod {2}}.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/09681ae056ad697750f77b115f9a01a043a52c39)
- The determinant of Jn is:
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:
![{\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}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ab3356c88e424c93f7edc2e2b529df1c31fb0709)
- The adjugate matrix of Jn is:
(where sgn is the sign of the permutation πk of k elements).
Relationships[edit]
- 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[edit]
References[edit]