Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form,
$$
J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix},
$$
where $s$ is the eigenvalue (with multiplicity $2$). My question: Is $J_2$ almost diagonalizable? I mean is it similar to
$$
\begin{bmatrix}s&\epsilon\\0&s\end{bmatrix}
$$
for every $\epsilon>0$ no matter how small? And what happens in higher dimensions, is every matrix similar to an arbitrarily close to diagonal matrix?
My guess is yes but I just can't find the similarity transformation. It could be something very simple.
Thanks in advance, all ideas welcome.
Best Answer
Yes, this is true and you gave the answer almost by yourself. The key is the Jordan normal form. To get $\epsilon$ instead of $1$, you just need to take the Jordan normal form of $\frac{1}{\epsilon}A$ and then multiply by $\epsilon$.