I have a few questions on matrix diagonalizability. Suppose that $A \in \mathbb{R}^{n \times n}$.
If $A$ is diagonalizable, we can find $P \in \mathbb{R}^{n \times n}$ such that $A = P\Lambda P^{-1}$. Does the definition of diagonalizability in $\mathbb{R}^{n \times n}$ mean that eigenvalues must be real as well? If we are allowed to have complex eigenvalues, can we write more matrices $A = P\Lambda P^{-1}$?
From the Wikipedia page, I know that the set of diagonalizable matrices on $\mathbb{C}^{n \times n}$ is dense. If $A \in \mathbb{R}^{n \times n}$ is non-diagonalizable, it can be approximated with a diagonalizable matrix $\widetilde{A} \in \mathbb{C}^{n \times n}$ arbitrarily closely. In general $\widetilde{A}$ may have complex eigenvectors. Is it possible to approximate $A$ arbitrarily closely with real eigenvectors and complex eigenvalues?