Is the Product of Two Diagonalizable Matrices Always Diagonalizable?

[Yankel]

Hello

I have a little question

if A and B are both invertible and diagonalizable matrices (from the same order), is A*B a diagonalizable matrix ? why ?

I have not got a clue...

thanks !

 

[Fernando Revilla]

if A and B are both invertible and diagonalizable matrices (from the same order), is A*B a diagonalizable matrix ? why ?

It is not true. Verify that $A=\begin{bmatrix}1&{\;\;1}\\{2}&{-1}\end{bmatrix}$ and $ B=\begin{bmatrix}{1}&{2}\\{2}&{1}\end{bmatrix}$ are invertible and diagonalzable matrices on $\mathbb{R}$, however $AB=\begin{bmatrix}{3}&{3}\\{0}&{3}\end{bmatrix}$ it is not diagonalizable.

 

[Richardhsu]

Hello

I have a little question

if A and B are both invertible and diagonalizable matrices (from the same order), is A*B a diagonalizable matrix ? why ?

I have not got a clue...

thanks !

Not always. But when A,B are interchangable, i.e., AB=BA, then AB IS diagonalizable since then A and B are simultaneously diagonalizable, i.e., we can find the same invertible matrix
S such that $A=S^{-1} D_1 S, B=S^{-1} D_2 S$, where $D_1,D_2$ are both diagonal matrices. Thus we have $AB=S^{-1}D S$ with $D=D_1D_2$ a diagonal matrix.