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.
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Transcription
Use the commutative law of multiplication to write 2 times 34 in a different way. Simplify both expressions to show that they have identical results. So once again, this commutative law just means that order doesn't matter. It sounds very fancy. Commutative law of multiplication. But all that says is that it doesn't matter whether we do 2 times 34 or whether we do 34 times 2. The order does not matter. We can commute the two terms. Both of these are going to get you the same exact answer. So let's try it out. What is 2 times 34? And we could write it like this, literally. You'll almost never see it written like this, but it is literally 2 times 34. Almost always people write the larger digit on top, or the digit with more digits, or the number with more digits on top. But let's do it this way. 4 times 2 is 8, and then we'll put a 0. 3 times 2 is 6, or you can view it as 30 times 2 is 60. Add them together. 8 plus 0 is 8. 6, bring it down. It's not being added to anything. You get 68. So 2 times 34 is 68. Now, if you do 34 times 2, 2 times 4 is 8, 2 times 3 is 6. That's why it's always nicer to write the number with more digits on top. It also is equal to 68. So it doesn't matter whether you have two groups of 34 or thirty-four groups of 2, in either case, you're going to have 68.
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
- Commutative property – Property of some mathematical operations
- Accumulator (cryptography)
References
- ^ a b Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.
- ^ 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.
This page was last edited on 5 July 2023, at 04:21