Inner automorphisms of a matrix algebra
= Inner automorphisms of a matrix algebra
{title2=$\operatorname{Aut}_L(M_d(L))=\operatorname{PGL}_d(L)$}
Every $L$-algebra automorphism of $M_d(L)$ is conjugation by an invertible matrix. For matrix units, the vectors $h(E_{i1})w$ with $0\ne w\in\operatorname{im}h(E_{11})$ form a basis in which $h$ acts in the usual way on all matrix units. Scalar conjugating matrices act trivially, giving $\operatorname{Aut}_L(M_d(L))=\operatorname{PGL}_d(L)$.