Quasi-commutative property explained

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

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:

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

f : X x Y \to X is said to be ^[2] if $$f\backslash left(f\backslash left(x,\; y\_1\backslash right),\; y\_2\backslash right)\; =\; f\backslash left(f\backslash left(x,\; y\_2\backslash right),\; y\_1\backslash right)\; \backslash qquad\; \backslash text\; x\; \backslash in\; X,\; \backslash ;\; y\_1,\; y\_2\; \backslash in\; Y.$$

If

f(x, y) is instead denoted by x \ast y then this can be rewritten as: $$(x\; \backslash ast\; y)\; \backslash ast\; y\_2\; =\; \backslash left(x\; \backslash ast\; y\_2\backslash right)\; \backslash ast\; y\; \backslash qquad\; \backslash text\; x\; \backslash in\; X,\; \backslash ;\; y,\; y\_2\; \backslash in\; Y.$$

Notes and References

1. 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.