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 $p$ 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:

$${\begin{aligned}xz&=zx\\yz&=zy\end{aligned}}$$

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 $f:X\times Y\to X$ is said to be quasi-commutative if

$$f\left(f\left(x,y_{1}\right),y_{2}\right)=f\left(f\left(x,y_{2}\right),y_{1}\right)\qquad {\text{ for all }}x\in X,\;y_{1},y_{2}\in Y.$$

If $f(x,y)$ is instead denoted by $x\ast y$ then this can be rewritten as:

$$(x\ast y)\ast y_{2}=\left(x\ast y_{2}\right)\ast y\qquad {\text{ for all }}x\in X,\;y,y_{2}\in Y.$$

See also

• Commutative property – Property of some mathematical operations
• Accumulator (cryptography)