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 $F(X)=SX{S}^{-1}$ 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.