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 [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] are said to have the commutative property whenever [math]\displaystyle{ pq = qp }[/math]
The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] [math]\displaystyle{ xy - yx = z }[/math]
satisfy the quasi-commutative property whenever [math]\displaystyle{ z }[/math] satisfies the following properties: [math]\displaystyle{ \begin{align} xz &= zx \\ yz &= zy \end{align} }[/math]
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, and z = iħ times the infinite unit matrix, where ħ is the reduced Planck constant.
Applied to functions
A function [math]\displaystyle{ f : X \times Y \to X }[/math] is said to be quasi-commutative[2] if [math]\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. }[/math]
If [math]\displaystyle{ f(x, y) }[/math] is instead denoted by [math]\displaystyle{ x \ast y }[/math] then this can be rewritten as: [math]\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. }[/math]
See also
- Commutative property – Property of some mathematical operations
- Accumulator (cryptography)
References
- ↑ 1.0 1.1 Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.
- ↑ Benaloh, J., & De Mare, M. (1994, January). One-way accumulators: A decentralized alternative to digital signatures. In Advances in Cryptology – EUROCRYPT’93 (pp. 274–285). Springer Berlin Heidelberg.
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