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 ${\displaystyle p}$ 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:

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

See also

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

References