Inner automorphisms of a matrix algebra (${\mathrm{Aut}}_L(M_d(L))={\mathrm{PGL}}_d(L)$)

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