Every automorphism of a full matrix algebra is inner
= Every automorphism of a full matrix algebra is inner
Every unital algebra <automorphism> $F:M_n(\mathbb C)\to M_n(\mathbb C)$ has the form $F(X)=SXS^{-1}$ for some invertible $S$. One proof transports the <matrix> units through $F$, chooses compatible bases for their rank-one images, and reconstructs the common similarity transformation.