- Source: Quasi-commutative property
In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions.
Applied to matrices
Two matrices
p
{\displaystyle p}
and
q
{\displaystyle q}
are said to have the commutative property whenever
p
q
=
q
p
{\displaystyle pq=qp}
The quasi-commutative property in matrices is defined as follows. Given two non-commutable matrices
x
{\displaystyle x}
and
y
{\displaystyle y}
x
y
−
y
x
=
z
{\displaystyle xy-yx=z}
satisfy the quasi-commutative property whenever
z
{\displaystyle z}
satisfies the following properties:
x
z
=
z
x
y
z
=
z
y
{\displaystyle {\begin{aligned}xz&=zx\\yz&=zy\end{aligned}}}
An example is found in the matrix mechanics introduced by Heisenberg as a version of quantum mechanics. In this mechanics, p and q are infinite matrices corresponding respectively to the momentum and position variables of a particle. These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite unit matrix, where ħ is the reduced Planck constant.
Applied to functions
A function
f
:
X
×
Y
→
X
{\displaystyle f:X\times Y\to X}
is said to be quasi-commutative if
f
(
f
(
x
,
y
1
)
,
y
2
)
=
f
(
f
(
x
,
y
2
)
,
y
1
)
for all
x
∈
X
,
y
1
,
y
2
∈
Y
.
{\displaystyle f\left(f\left(x,y_{1}\right),y_{2}\right)=f\left(f\left(x,y_{2}\right),y_{1}\right)\qquad {\text{ for all }}x\in X,\;y_{1},y_{2}\in Y.}
If
f
(
x
,
y
)
{\displaystyle f(x,y)}
is instead denoted by
x
∗
y
{\displaystyle x\ast y}
then this can be rewritten as:
(
x
∗
y
)
∗
y
2
=
(
x
∗
y
2
)
∗
y
for all
x
∈
X
,
y
,
y
2
∈
Y
.
{\displaystyle (x\ast y)\ast y_{2}=\left(x\ast y_{2}\right)\ast y\qquad {\text{ for all }}x\in X,\;y,y_{2}\in Y.}
See also
Commutative property – Property of some mathematical operations
Accumulator (cryptography)
References
Kata Kunci Pencarian:
- Quasi-commutative property
- Commutative property
- Coherent sheaf
- Quasi-free algebra
- Associative algebra
- Noncommutative geometry
- Local ring
- Spectrum of a ring
- Ring (mathematics)
- Quasi-Frobenius ring
