1. ## [SOLVED] Permutation Matrices

Hey all, this is one of my practice questions and I have no idea where to start.
Find a 5 by 5 permutation matrix P which has 1 in its (5,4)-place and so that the smallest power of P which gives the identity matrix is P6 .
(Hint: Consider "combining" a 2 by 2 permutation matrix with a 3 by 3 one.)

There are several correct different solutions.

The question has been designed so that there should be a block diagonal permutation matrix solution.

I haven't actually encountered permutation matrices before, but I've looked up the basics (what they do, how to apply them).

We're looking for a permutation which has order 6, the hint given to us suggests using a permutation of order 2 and a permutation of order 3 together (which will after all give a permutation of order lcm(3,2) = 6)

Now, how can we create such a permutation, given the requirement that it will shift 4->5

3. Do you mean permutations as in {0,0,1} to {0,1,0} to.. etc? I'm not quite sure how you'd combine two permutations of two different orders, something like

Perm[{0,0,1}] combined with Perm[{0,1}]?

Which I guess would give {0,0,1,0,1} and {0,0,1,1,0}, amongst other permutations, but they're of order 5. Can you explain a little more?

*edit* I should really add that this was designed as a 'revision problem', but we haven't been taught such a thing before (I guess it was assumed prior knowledge). So layman's terms are fine

4. I ended up solving the problem by writing a brute force algorithm in Mathematica, but I'd still like to know how to do it by hand. I can post the code if anybody is interested, but it is fairly straight forward.

5. Well, what is the permutation matrix for the element of $\displaystyle S_5$ which is (123)(45)

6. Originally Posted by SimonM
Well, what is the permutation matrix for the element of $\displaystyle S_5$ which is (123)(45)
The matrix which changes the vector (1,2,3,4,5) -> (1,2,3,5,4) would be

|1 0 0 0 0|
|0 1 0 0 0|
|0 0 1 0 0|
|0 0 0 0 1|
|0 0 0 1 0|

Which has a 1 in the element (5,4). Is this what you meant?

Or a permutation matrix which changes (123)(45) -> (45)(123) would be

|0 0 0 0 1|
|0 0 0 1 0|
|1 0 0 0 0|
|0 1 0 0 0|
|0 0 1 0 0|

7. You've got the element (45) as your first matrix, but we want (123)(45) (the matrix which permutes cyclically 1,2,3 at the same time)

8. Okay,

|0 1 0 0 0|
|0 0 1 0 0|
|1 0 0 0 0|
|0 0 0 0 1|
|0 0 0 1 0|

Should cycle (45) as well as (123) cyclically.