Conflicting theorems regarding eigenvalues & matrix diagonalisability

Question

I have the two following theorems:

• Matrix $A$ is diagonalisable if and only if the algebraic multiplicity equals the geometric multiplicity for every eigenvalue and all its eigenvalues are inside its field.
• If matrix $A$ has a single eigenvalue it is diagonalisable if and only if it is a scalar matrix.

I have the following matrix above $\mathbb{R}$:

I calculated its characteristic polynomial to be $\lambda^3 - 8$, which in turn means $\lambda = 2$ is its only eigenvalue, which leads to a conflict between the theorems — the first theorem says it's diagonalisable, second does not.

Why?

Answer 1

Both theorems imply your example is not diagonalizable. There are complex roots of $\lambda^3-8=0$, so not all eigenvalues are in the real field.