Fields over which matrix is diagonalizable

Question

I recently came across diagonalizations of certain special matricies in my Linear Algebra lecture (real symmetric, self-adjoint, orthogonal, etc), and have a couple questions about knowing which field(s) a matrix is diagonalizable over:

1. If a matrix with complex entries is diagonalizable, is it only diagonalizable over $\mathbb{C}$, or are there such matricies that are also diagonalizable over $\mathbb{R}$?
2. If a matrix with complex entries has only real eigenvalues (which is the case for self-adjoint matricies), is it diagonalizable only over $\mathbb{C}$?
3. If a matrix with real entries is diagonalizable, is it always diagonalizable over $\mathbb{R}$ and not just $\mathbb{C}$?

Answer 1

1. The question does not make much sense. If the matrice has non-real entries, what does it mean to be diagonalizable over $\mathbb{R}$ ?

2. No, the matrix may be nondiagonalizable over $\mathbb{C}$. For example $$A=\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}$$

is nilpotent, so its only eigenvalue is $0$ which is real, but $A$ is not diagonalizable (since it is nilpotent and nonzero).

3. If a real matrix is diagonalizable over $\mathbb{R}$, then obviously it is also diagonalizable over $\mathbb{C}$. The converse however is not true. Take for example $$A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

It is diagonalizable over $\mathbb{C}$, but not over $\mathbb{R}$.