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.{{citation|title=Matrix Analysis|first1=Roger A.|last1=Horn|first2=Charles R.|last2=Johnson|edition=2nd|publisher=Cambridge University Press|year=2012|isbn=978-1-139-78888-5 |chapter=§0.9.5.1 n-by-n reversal matrix |page=33|chapter-url=https://books.google.com/books?id=O7sgAwAAQBAJ&pg=PA33}}.
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}
Definition
If {{mvar|J}} is an {{math|n × n}} exchange matrix, then the elements of {{mvar|J}} are
1, & i + j = n + 1 \\
0, & i + j \ne 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 {{mvar|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, {{mvar|Jn}} is an involutory matrix; that is,
J_n^{-1} = J_n.
- The trace of {{mvar|Jn}} is 1 if {{mvar|n}} is odd and 0 if {{mvar|n}} is even. In other words:
\operatorname{tr}(J_n) = \frac{1-(-1)^n}{2} = n\bmod 2.
- The determinant of {{mvar|Jn}} is:
\det(J_n) = (-1)^{\lfloor n/2\rfloor} = (-1)^\frac{n(n-1)}{2}
As a function of {{mvar|n}}, it has period 4, giving 1, 1, −1, −1 when {{mvar|n}} is congruent modulo 4 to 0, 1, 2, and 3 respectively.
- The characteristic polynomial of {{mvar|Jn}} 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 ) and -1 (with multiplicity ).
- The adjugate matrix of {{mvar|Jn}} is:
\operatorname{adj}(J_n) = \sgn(\pi_n) J_n.
(where {{math|sgn}} is the sign of the permutation {{mvar|π{{sub|k}}}} of {{mvar|k}} elements).
Relationships
- An exchange matrix is the simplest anti-diagonal matrix.
- Any matrix {{mvar|A}} satisfying the condition {{math|1=AJ = JA}} is said to be centrosymmetric.
- Any matrix {{mvar|A}} satisfying the condition {{math|1=AJ = JAT}} is said to be persymmetric.
- Symmetric matrices {{mvar|A}} that satisfy the condition {{math|1=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)