According to Wikipedia ("Diagonal Matrix", 8-Dec-2019, https://en.wikipedia.org/wiki/Diagonal_matrix): "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices." The diagonal matrices are mentioned in [Lang] p. 576, but without giving them a dedicated definition. Furthermore, "A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple of the identity matrix . Its effect on a vector is scalar multiplication by [see scmatscm!]". The scalar multiples of the identity matrix are mentioned in [Lang] p. 504, but without giving them a special name.
The main results of this subsection are the definitions of the sets of diagonal and scalar matrices (df-dmat and df-scmat), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng), that the scalar matrices are a subring of the ring of square matrices (scmatsrng), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng).