$T:\mathbb C^n \to \mathbb C^n$ is linear and $\ker(T-aI)=\ker(T-aI)^n , \forall a\in \mathbb C$ , then $T$ diagonalizable ?

Question

If $T:\mathbb C^n \to \mathbb C^n$ is a linear transform such that $\ker(T-aI)=\ker(T-aI)^n , \forall a\in \mathbb C$ , then is $T$ diagonalizable ?

Answer 1

Note that for Any $a \in \Bbb C$, $\dim\ker(T - aI)$ is the geometric multiplicity of $a$ (as an eigenvalue of $T$), whereas $\dim\ker(T-aI)^n$ will always be the algebraic multiplicity of $a$.