On semigroup of quaternionic matrices

In this paper, we derived some algebraic properties of matrices over the quaternions. A ∗-isomorphism between the [Formula: see text] matrices over the quaternions [Formula: see text], and a subset of the [Formula: see text] matrices [Formula: see text] over the complex numbers is established. A characterization of idempotent matrices in terms of block square matrices of order 2 is provided. Following this, we characterized idempotents in matrices of quaternions in terms of its complex representations. A study of ∗-semigroup structure of matrices over quaternions is also done in this work.