Quasi-commutative property

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In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in certain specific applications with various definitions.

Applied to matrices[edit]

Two matrices p and q are said to have the commutative property whenever

pq = qp

The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices x and y

 xy - yx = z

satisfy the quasi-commutative property whenever z satisfies the following properties:

 xz = zx
 yz = zy

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[edit]

A function f, defined as follows:

f: X \times Y \rightarrow X

is said to be quasi-commutative[2] if for all x \in X and for all y_1, y_2 \in Y,

f(f(x,y_1),y_2) = f(f(x,y_2),y_1)

See also[edit]


  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.