Permutation Matrices

1. Apr 3, 2013

dylanhouse

1. The problem statement, all variables and given/known data

Supposing P is a permutation matrix, I have to show that PT(I+P) = (I+P)T. Is there any general form of a permutation matrix I should use here as permutation matrices of a dimension can come in various forms.

2. Relevant equations

3. The attempt at a solution

I did this letting P = [0, 1| and it did indeed work out fine.
|1, 0]

2. Apr 3, 2013

ArcanaNoir

Doesn't P have to be a square matrix? Or maybe I'm just not familiar with your notation?

3. Apr 3, 2013

dylanhouse

Yes, it does have to be square. I know it must also have a single 1 in each row and column, the rest zeros. But this can happen in multiple ways correct? So is there not a generalized form for a permutation matrix?

ie. [1 0|
|0 1]

OR

[0 1|
|1 0]

4. Apr 3, 2013

micromass

Staff Emeritus
What is the definition of a "permutation matrix"?

5. Apr 3, 2013

dylanhouse

It is a matrix created from the identity by arranging rows and columns. It has a single 1 in each row and column; the rest are zeros.

6. Apr 3, 2013

micromass

Staff Emeritus
OK. Then that's the only thing you can use. You can't pick a special form of $P$ and prove it for that. You need to prove it for all possible $P$.

That said, are you familiar with elementary row and column transformations? This can help you. Why? Because any permutation matrix can be made from the identity matrix by just exchanging a few columns and rows.

7. Apr 3, 2013

dylanhouse

I know that to get P I can multiply I by an elementary matrix. I understand the concept, but am unsure how I am supposed to go about proving this question. The transpose of the permutation will always just be the permutation.. correct?

8. Apr 3, 2013

micromass

Staff Emeritus
The idea is to prove the equation $P^T(I+P) = (I+P)^T$ first for elementary matrices that switch a row or a column. Then you should only show that if two matrices $P$ and $Q$ satisfy the equation, then so does their product. I claim that this shows that the equation holds for each permutation matrix.