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



  1. ^ a b Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.
  2. ^ 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.