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: $$\begin{array}{cc}xz& =zx\\ yz& =zy\end{array}$$

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 Template:Visible anchor if $$f(f(x,y_1),y_2)=f(f(x,y_2),y_1)\text{ for all }x\in X,y_1,y_2\in Y.$$

If $f(x,y)$ is instead denoted by $x\ast y$ then this can be rewritten as: $$(x\ast y)\ast y_2=(x\ast y_2)\ast y\text{ for all }x\in X,y,y_2\in Y.$$

See also

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