- Source: Exchange matrix
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.
J
2
=
(
0
1
1
0
)
J
3
=
(
0
0
1
0
1
0
1
0
0
)
⋮
J
n
=
(
0
0
⋯
0
1
0
0
⋯
1
0
⋮
⋮
⋅
⋅
j
˙
⋮
⋮
0
1
⋯
0
0
1
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⋯
0
0
)
{\displaystyle {\begin{aligned}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{aligned}}}
Definition
If J is an n × n exchange matrix, then the elements of J are
J
i
,
j
=
{
1
,
i
+
j
=
n
+
1
0
,
i
+
j
≠
n
+
1
{\displaystyle J_{i,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}}
Properties
Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,
(
0
0
1
0
1
0
1
0
0
)
(
1
2
3
4
5
6
7
8
9
)
=
(
7
8
9
4
5
6
1
2
3
)
.
{\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}}.}
Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,
(
1
2
3
4
5
6
7
8
9
)
(
0
0
1
0
1
0
1
0
0
)
=
(
3
2
1
6
5
4
9
8
7
)
.
{\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}}.}
Exchange matrices are symmetric; that is:
J
n
T
=
J
n
.
{\displaystyle J_{n}^{\mathsf {T}}=J_{n}.}
For any integer k:
J
n
k
=
{
I
if
k
is even,
J
n
if
k
is odd.
{\displaystyle 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, Jn is an involutory matrix; that is,
J
n
−
1
=
J
n
.
{\displaystyle J_{n}^{-1}=J_{n}.}
The trace of Jn is 1 if n is odd and 0 if n is even. In other words:
tr
(
J
n
)
=
n
mod
2
.
{\displaystyle \operatorname {tr} (J_{n})=n{\bmod {2}}.}
The determinant of Jn is:
det
(
J
n
)
=
(
−
1
)
n
(
n
−
1
)
2
{\displaystyle \det(J_{n})=(-1)^{\frac {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
(
λ
I
−
J
n
)
=
{
[
(
λ
+
1
)
(
λ
−
1
)
]
n
2
if
n
is even,
(
λ
−
1
)
n
+
1
2
(
λ
+
1
)
n
−
1
2
if
n
is odd.
{\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}}}
The adjugate matrix of Jn is:
adj
(
J
n
)
=
sgn
(
π
n
)
J
n
.
{\displaystyle \operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}.}
(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
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