Solving Diagonal Matrix Homework: AB=BA

[arshavin]

Homework Statement

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

Homework Equations

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..

 

[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.