Prove: matrix A is diagonalizable iff exp(A) is diagonalizble I need to prove: matrix A is diagonalizable iff $\exp(A)$ is diagonalizble. exp means exponent function. 
I know to prove that if $A$ is diagonalizable so $\exp(A)$ is diagonalizable, but have a problem with the other side. 
can I write $P^{-1}.\exp(A).P=D$ (since $\exp(A)$ is diagonalizable  ) and operate log on both sides of the equation? if yes I'm done, or any other hints?
 A: Since you already know that "$A$ is diagonalisable" implies that "$\exp(A)$ is diagonalisable", we show that if $A$ is not diagonalisable, then neither is $\exp(A)$. 

$A$ is not diagonalisable $\Rightarrow$ $\exp(A)$ is not diagonalisable.

Let $A$ be non-diagonalisable and let $A=XJX^{-1}$ be its Jordan form, where $J$ is a direct sum of elementary Jordan blocks: $J=\oplus_{i=1}^l J_i$. The function $\exp(A)$ can then be written in the form $\exp(A)=X\exp(J)X^{-1}$, where $\exp(J)$ is again a direct sum of exponentials of elementary Jordan blocks: $\exp(J)=\oplus_{i=1}^l \exp(J_i)$. Since $A$ is non-diagonalisable and so is $J$, $J$ contains at least one non-trivial Jordan block
$$
\tilde{J}=\lambda I + N\in\mathbb{C}^{k\times k}, \quad k\geq 2,
$$
where $\lambda\in\mathbb{C}$ is an eigenvalue of $A$ and $N$ is the nilpotent matrix
$$
N=\begin{bmatrix}
0 & 1 &        &        &   \\
  & 0 & 1      &        &   \\
  &   & \ddots & \ddots &   \\
  &   &        & 0      & 1 \\
  &   &        &        & 0
\end{bmatrix}.
$$
If we show that $\exp(\tilde{J})$ of such a non-trivial block is non-diagonalisable, we are done.
Looking again here how $\exp(J)$ looks like, we see that
$$
\exp(\tilde{J})=\exp(\lambda)\exp(N)
=\exp(\lambda)\sum_{i=0}^{k-1}\frac{1}{i!}N^i.
$$
Note that $1$ is the only eigenvalue of the upper triangular matrix $\exp(N)$.
So $\exp(N)$ is diagonalisable iff the kernel of the $k\times k$ matrix $\exp(N)-I$ is $\mathbb{C}^{k}$. This is, however, not the case as $\exp(N)-I$ is not a zero matrix. Therefore, $\exp(N)$ and consequently $\exp(\tilde{J})$ and consequently $\exp(A)$ is not diagonalisable.
A: Try the Taylor series expansion for the exponent. For small $t$, $e^{tA} \approx I + tA$
