Products of hermitian or skew-Hermitian matrices over division rings with involution

Let D be a division ring with involution. In this paper, we investigate conditions under which a square matrix in Mn(D) can be expressed as a product of finitely many Hermitian or skew-Hermitian matrices. Among other results, we prove that every non-invertible matrix in Mn(D) can be written as a product of four matrices, each of which is Hermitian or skew-Hermitian, in any prescribed pattern. We also show that every matrix in SLn(D) can be expressed as a product of finitely many Hermitian or skew-Hermitian matrices, and we derive an upper bound for the required number. Our results improve several known theorems, including those in the recent work [Citation3].