If matrix is diagonalizable, eigenvalue? 
Let $A$ be an $n \times n$ matrix and suppose $A$ is diagonalizable and the only eigenvalue is $\lambda = k$, what can you say about matrix $D$ where $A = P^{-1} D P$, for invertible matrix $P$.

So if the only eigenvalue of $A$ is $\lambda = k$, what can I say about $D$? 
I know that $D$ is a diagonal matrix, but is it necessarily true that $D = \text{diag } (k, k, ... , k)$ ?

Yes, it is. One way to see this is that eigenvalues are invariant under conjugation. This is a fancy way to say that if 
$$
A=PDP^{-1}
$$
then $A$ and $D$ have the same eigenvalues. 
Now, what are the eigenvalues of a diagonal (or upper triangular) matrix? 
Edit: Proof of fact mentioned in comments:
suppose 
$$
\lambda I=PAP^{-1}
$$
for some change of basis matrix $P$. Then 
$$
A=P^{-1}\lambda IP=\lambda P^{-1}P=\lambda I
$$
