Diagonal matrix

1. Oct 11, 2008

arshavin

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

Suppose the matrix A , B are diagonalizable and have the same eigenvectors. Show AB=BA

2. Relevant equations

3. The attempt at a solution

There exists a matrix P s.t. (P^-1)AP=(P^-1)BP

I played around with this, and could not get anywhere..

2. Oct 11, 2008

Dick

If A is diagonalizable then there is a basis where it's a diagonal matrix. Then all of the basis vectors are eigenvectors. If B has the same eigenvectors then it's also a diagonal matrix in the same basis. Now you just have to show two diagonal matrices commute.