TITLE: Density of Diagonalizable matrices
QUESTION [8 upvotes]: Suppose that the matrix $A$ is not diagonalizable. Show that for all $\epsilon>0$, there exists a diagonalizable matrix $A_{\epsilon}$ such that $||A-A_{\epsilon}||_{2}<\epsilon$. Can anyone help me prove this or give a hint?

REPLY [3 votes]: The claim in the OP is only correct if the matrix elements lie in an algebraically closed field. For example, the set of diagonalizable matrices over $\mathbb{R}$ is not a dense subset of the space of all matrices over $\mathbb{R}$ (under the standard vector topology), as proven here.