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

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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

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A function is said to be quasi-commutative[2] if

If is instead denoted by then this can be rewritten as:

See also

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References

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  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.