# Thread: [SOLVED] Permutation Matrices

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).

2. Rather than thinking about permutation matrices, think about permutations.

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 $S_5$ which is (123)(45)

6. Originally Posted by SimonM
Well, what is the permutation matrix for the element of $S_5$ 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.

9. And that's my answer! Thanks for your patience.

10. Excellent. You might also want to find all the other permutation matrices which work (as an extension problem)