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 and are said to have the commutative property whenever
The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices and
satisfy the quasi-commutative property whenever satisfies the following properties:
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 is said to be quasi-commutative[2] if
If is instead denoted by then this can be rewritten as:
See also[]
- Commutative property – Property of a mathematical operation
- Accumulator (cryptography)
References[]
- ^ a b 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.
- Mathematical relations
- Properties of binary operations