If A and B are both diagonalizable, is AB also diagonalizable?

Question

Here are my steps:

$PAP^{-1}PBP^{-1}=PABP^{-1}=AB$, since both are diaglonalizale, the equation is established, is it correct?

Not really... it is not "the same $P$." – Clement C. Dec 16 '15 at 16:44 — Clement C., Dec 16 '15 at 16:44
The answer is no, and an example can be found here. – Pierre-Guy Plamondon Dec 16 '15 at 16:46 — Pierre-Guy Plamondon, Dec 16 '15 at 16:46

score 1 · Answer 1 · answered Dec 16 '15 at 16:55

1

The result is not correct as said in the comments.

However it is true when $A$ and $B$ commute. This last result can be proven as the eigenspace of one matrix is a stable subspace of the other matrix.

answered Dec 16 '15 at 16:55

1 Answers1