# Quasi-commutative property

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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 and q are said to have the commutative property whenever

$pq=qp$ The quasi-commutative property in matrices is defined as follows. Given two non-commutable matrices x and y

$xy-yx=z$ satisfy the quasi-commutative property whenever z satisfies the following properties:

$xz=zx$ $yz=zy$ 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, defined as follows:

$f:X\times Y\rightarrow X$ is said to be quasi-commutative if for all $x\in X$ and for all $y_{1},y_{2}\in Y$ ,

$f(f(x,y_{1}),y_{2})=f(f(x,y_{2}),y_{1})$ 