Suppose $A\in M_n(\Bbb C)$ satisfies $A^6-A^3+I=O$. Prove that if a linear transformation $T:M_n(\Bbb C)\rightarrow M_n(\Bbb C)$ is given by $T(B)=AB$, then $T$ is diagonalizable.
How to prove it? Is it related to minimal polynomial? I have no idea..
The polynomial $f(x)=x^6-x^3+1$ has six distinct complex roots, and annihilates $A$. Hence the minimal polynomial of $A$ has distinct complex roots (we don't know how many, but it it is at most six). Hence $A$ is diagonalizable, because having only linear terms in a minimal polynomial is a characterization of diagonalizability.
Addendum: $f(x)(x^3+1)=x^9+1$, which has nine distinct complex roots (complex ninth roots of $-1$).