The diagonalizable matrices are not dense in the square real matrices

Question

Suppose that $n \ge 2$. How to prove that the set $\mathcal D \subset M_n(\mathbb R)$ of the diagonalizable real matrices is not dense in $M_n(\mathbb R)$?

Answer 1

Consider the matrix :

$$R=\pmatrix{0 & -1\cr 1 & 0}\in M_2(\mathbb{R})$$

which is not diagonalizable since its characteristic polynomial $X^2+1$ does not split in $\mathbb{R}[X]$.

Suppose there exists a sequence $(D_n)$ of diagonalizable matrices in $M_2(\mathbb{R})$, which converges to $R$.

For every $n$, the characteristic polynomial of $D_n$ has nonnegative discriminant and by continuity of the determinant, it should be the same for $R$, but this is not the case.

This proves that the set $\mathcal{D}_2$ of all diagonalizable matrices in $M_2(\mathbb{R})$ is not dense in $M_2(\mathbb{R})$.