Elegant way to prove ##AB^{n}## commute

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In summary, there are several elegant approaches to proving that ##AB^{n}## commute, including using mathematical induction. Commuting in mathematical terms refers to two elements multiplying in either order with the same result. It is not necessary to prove that ##AB^{n}## commute for all values of n, but it is necessary to show that the statement holds for all possible values. Understanding when two elements commute can have real-world applications in fields such as physics and engineering. There are alternative methods for proving that ##AB^{n}## commute, such as using matrix properties or the definition of commutativity.
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If ##A## and ##B## are two matrices which commute, then probably ##AB^{n} = B^{n}A## too.

Now, I could probably do all the work of proving this with induction, but I feel like there should be a more elegant way to prove this, though I don't know exactly how to avoid writing ellipses. If I start with ##AB^{n} = A \prod_{k=1}^{n}B = AB...B = BAB... = BBAB...B -> = BB...BA ...## and we get this sort of iterative commutation argument, though I'm not sure how to formalize it.
 
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Induction is elegant, especially in cases like this!
 
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1. How do you define an "elegant" proof?

An elegant proof is one that is concise, logical, and easy to follow. It often involves using creative and efficient techniques to arrive at a solution.

2. What is the significance of proving that ##AB^{n}## commute?

Proving that ##AB^{n}## commute is important in various fields of mathematics and science, as it allows for simplification and manipulation of equations and matrices. It also helps to establish relationships between different mathematical concepts.

3. What are some common techniques used to prove that ##AB^{n}## commute?

Some common techniques used to prove that ##AB^{n}## commute include induction, direct proof, and the use of commutativity properties of matrices and operators.

4. Can you provide an example of an elegant proof for ##AB^{n}## commute?

Sure, one elegant proof for ##AB^{n}## commute involves using induction and the commutativity property of matrices. By proving the base case (##n=1##) and then showing that if ##AB^{k}## commute, then ##AB^{k+1}## also commute, we can establish that ##AB^{n}## commute for all natural numbers ##n##.

5. Are there any real-world applications for proving that ##AB^{n}## commute?

Yes, there are many real-world applications for proving that ##AB^{n}## commute. For example, in physics, this proof can be used to simplify equations involving operators and matrices, making calculations more efficient. In computer science, it can be used to optimize algorithms and data structures. It also has applications in engineering, economics, and other fields.

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