In mathematics, the quasi-commutative property is an extension or generalization of the general

commutative property In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

. This property is used in specific applications with various definitions.

Applied to matrices

Two

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

p

and

q

are said to have the

whenever

pq = qp

The quasi-commutative property in matrices is definedNeal H. McCoy
On quasi-commutative matrices. ''Transactions of the American Mathematical Society, 36''(2), 327–340
as follows. Given two non-commutable matrices

x

and

y

xy - yx = z

satisfy the quasi-commutative property whenever

z

satisfies the following properties:

\begin
  xz &= zx \\
  yz &= zy
\end

An example is found in the

matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...

introduced by

Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...

as a version of

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

. In this mechanics, ''p'' and ''q'' are infinite matrices corresponding respectively to the momentum and position variables of a particle. These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite

unit matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...

, where ħ is the

reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...

Applied to functions

A function

f : X \times Y \to X

is said to be 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. if

f\left(f\left(x, y_1\right), y_2\right) = f\left(f\left(x, y_2\right), y_1\right) \qquad \text x \in X, \; y_1, y_2 \in Y.

f(x, y)

is instead denoted by

x \ast y

then this can be rewritten as:

(x \ast y) \ast y_2 = \left(x \ast y_2\right) \ast y \qquad \text x \in X, \; y, y_2 \in Y.

References

{{reflist, 30em Mathematical relations Properties of binary operations

Applied to matrices

Applied to functions

See also

References