Real-diagonalisable matrices: neither open nor closed

Idea. Diagonalisable = no non-trivial Jordan block; a Jordan block can be created by an arbitrarily small perturbation (not open) and is a limit of diagonalisable matrices with distinct eigenvalues (not closed).

Step 1: not open ($n=2$). $I_2$ is diagonalisable. The matrices $B_\varepsilon=\begin{pmatrix}1&\varepsilon\\0&1\end{pmatrix}$ ($\varepsilon\ne0$) satisfy $B_\varepsilon-I=\varepsilon E_{12}\ne0$ but $(B_\varepsilon-I)^2=0$; the minimal polynomial is $(t-1)^2$, which has a repeated root, so $B_\varepsilon$ is not diagonalisable. Yet $B_\varepsilon\to I$.

Step 2: not closed ($n=2$). For $\varepsilon>0$, $C_\varepsilon=\begin{pmatrix}1&1\\0&1+\varepsilon\end{pmatrix}$ has two distinct real eigenvalues $1,1+\varepsilon$ and is therefore diagonalisable. As $\varepsilon\to0$, $C_\varepsilon\to C_0=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, which is a non-trivial Jordan block (minimal polynomial $(t-1)^2$) and is *not* diagonalisable.

Step 3: all $n\ge2$. Put $B_\varepsilon\oplus I_{n-2}$, $C_\varepsilon\oplus I_{n-2}$ (block diagonal). A block-diagonal matrix is diagonalisable iff each block is, so the same sequences show non-openness and non-closedness. (If one prefers eigenvalues to stay distinct in Step 2, use $C_\varepsilon\oplus\operatorname{diag}(2,3,\dots,n-1)$; this is not needed.)

Remark.
- In Part 7 we shall see the correct statements: the *interior* is the set of matrices with $n$ distinct real eigenvalues and the *closure* is the set of matrices whose characteristic polynomial splits over $\mathbb{R}$ (two problems below).
- Over $\mathbb{C}$ the situation is better: the set of matrices with distinct eigenvalues is open and dense, so diagonalisable matrices are dense (but still not open).