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Quasi-commutative property

Summary

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 ${\displaystyle p}$  and ${\displaystyle q}$  are said to have the commutative property whenever ${\displaystyle pq=qp}$

The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices ${\displaystyle x}$  and ${\displaystyle y}$  ${\displaystyle xy-yx=z}$

satisfy the quasi-commutative property whenever ${\displaystyle z}$  satisfies the following properties: {\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.

Applied to functions

A function ${\displaystyle f:X\times Y\to X}$  is said to be quasi-commutative[2] if ${\displaystyle f\left(f\left(x,y_{1}\right),y_{2}\right)=f\left(f\left(x,y_{2}\right),y_{1}\right)\qquad {\text{ for all }}x\in X,\;y_{1},y_{2}\in Y.}$

If ${\displaystyle f(x,y)}$  is instead denoted by ${\displaystyle x\ast y}$  then this can be rewritten as: ${\displaystyle (x\ast y)\ast y_{2}=\left(x\ast y_{2}\right)\ast y\qquad {\text{ for all }}x\in X,\;y,y_{2}\in Y.}$