# Math Help - permutation matrix

1. ## permutation matrix

Prove that permutation matrices can be written as the product of transpositions.

So geometrically, we can represent transpositions as lines between two "poles" or sticks. And completing the cycle defines a permutation?

2. Originally Posted by manjohn12
Prove that permutation matrices can be written as the product of transpositions.

So geometrically, we can represent transpositions as lines between two "poles" or sticks. And completing the cycle defines a permutation?
for any $\alpha, \beta \in S_n,$ let $P_{\alpha}, P_{\beta}$ be the corresponding permutation matrices. then $P_{\alpha}P_{\beta}=P_{\beta \alpha }.$ now use this fact that every $\sigma \in S_n$ is a product of transpositions.

3. because every transposition matrix is a permutation?

4. Originally Posted by manjohn12

because every transposition matrix is a permutation?
ok, how do you basically define a transposition matrix?

5. Originally Posted by NonCommAlg
for any $\alpha, \beta \in S_n,$ let $P_{\alpha}, P_{\beta}$ be the corresponding permutation matrices. then $P_{\alpha}P_{\beta}=P_{\beta \alpha }.$ now use this fact that every $\sigma \in S_n$ is a product of transpositions.

Is there not a neat way of showing the result for permutation matrices without going back into $S_n$?

6. Originally Posted by NonCommAlg
ok, how do you basically define a transposition matrix?
by interchanging two rows of the identity matrix

7. Originally Posted by manjohn12

by interchanging two rows of the identity matrix
right! and when you interchange the rows $i,j,$ you exactly get the permutation matrix $P_{\sigma},$ where $\sigma=(i \ j).$ so, every transposition matrix is a permutation.