Solving Permutation Matrices: Show PT(I+P)=(I+P)T

[dylanhouse]

Homework Statement

Supposing P is a permutation matrix, I have to show that P^T(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.

Homework Equations

The Attempt at a Solution

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

 

[ArcanaNoir]

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

 

[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]

 

[micromass]

What is the definition of a "permutation matrix"?

 

[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.

 

[micromass]

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.

 

[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?

 

[micromass]

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.