# The diagonalisation of the two matrices

My problem: If I have two matrices $A$ and $B$ and $A$ is diagonalisable and $B$ isn't, can $AB$ be diagonalisable?

My solution: Those matrices are diagonisable if there is a regular matrix $Q$ that satisfies:

$AB=QD_AQ^{-1}QD_BQ^{-1}$

where $D_A, D_B$ are the digonal matrices with the eigenvalues of matrices $A$ and $B$ on their diagonal.

• I don't understand how $Q$ looks like, it's obviosly a matrix containing the eigenvectors, $A$'s eigenvectors?
• Answer: $AB$ may or may not be diagonalisable. Example: $A=0$ and $B$ any non-diagonalisable matrix. Then $AB=0$ is diagonalisable. However, for $A=I_n$, and $B$ as before, $AB=B$ isn't . – Dietrich Burde Mar 5 '17 at 20:17
• It's not clear what your solution has to do with the problem. Also, how did you come across this problem? It seems oddly open ended. – Omnomnomnom Mar 5 '17 at 20:40