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 $D = \text{diag } (k, k, … , k)$ ?
Best Answer
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 $$ A=P^{-1}\lambda IP=\lambda P^{-1}P=\lambda I $$