# Math Help - [SOLVED] Diagonalization question 2

1. ## [SOLVED] Diagonalization question 2

Let A and B be unkown diagonalizable matrices. If A and B have exactly the same eigenvectors as each other, then show that AB = BA.

2. Originally Posted by jkeatin
Let A and B be unkown diagonalizable matrices. If A and B have exactly the same eigenvectors as each other, then show that AB = BA.
$A=P_1^{-1}D_1P_1, \ \ B=P_2^{-1}D_2P_2,$ where $D_1,D_2$ are diagonal and $P_1,P_2$ are matrices with the eigenvectors of $A,B$ as their columns repectively. so $P_1=P_2$ and since diagonal matrices commute

with each other, we'll have: $AB=P_1^{-1}D_1D_2P_1^{-1}=P_1^{-1}D_2D_1P_1=BA. \ \Box$

3. ok i got ya, thanks for the help man