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 $\displaystyle \alpha, \beta \in S_n,$ let $\displaystyle P_{\alpha}, P_{\beta}$ be the corresponding permutation matrices. then $\displaystyle P_{\alpha}P_{\beta}=P_{\beta \alpha }.$ now use this fact that every $\displaystyle \sigma \in S_n$ 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 $\displaystyle \alpha, \beta \in S_n,$ let $\displaystyle P_{\alpha}, P_{\beta}$ be the corresponding permutation matrices. then $\displaystyle P_{\alpha}P_{\beta}=P_{\beta \alpha }.$ now use this fact that every $\displaystyle \sigma \in S_n$ is a product of transpositions.

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

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 $\displaystyle i,j,$ you exactly get the permutation matrix $\displaystyle P_{\sigma},$ where $\displaystyle \sigma=(i \ j).$ so, every transposition matrix is a permutation.