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.
Two matrices
p
q
are said to have the commutative property whenever
The quasi-commutative property in matrices is defined[1] as follows.
Given two non-commutable matrices
satisfy the quasi-commutative property whenever
satisfies the following properties:
= z x
{\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.
[1] 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.
A function
is said to be quasi-commutative[2] if
is instead denoted by
then this can be rewritten as: