Thread: 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?

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 let be the corresponding permutation matrices. then now use this fact that every is a product of transpositions.

because every transposition matrix is a permutation?

Originally Posted by manjohn12

because every transposition matrix is a permutation?

ok, how do you basically define a transposition matrix?

Originally Posted by NonCommAlg

for any let be the corresponding permutation matrices. then now use this fact that every is a product of transpositions.

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

Originally Posted by NonCommAlg

ok, how do you basically define a transposition matrix?

by interchanging two rows of the identity matrix

Originally Posted by manjohn12

by interchanging two rows of the identity matrix

right! and when you interchange the rows you exactly get the permutation matrix where so, every transposition matrix is a permutation.