Abstract
This paper investigates conditions under which a matrix over an involutory division ring can be expressed as a product of Hermitian matrices, and estimates the minimal length of such factorizations. Let D be a division ring with center F and involution ⁎. Among others, we prove the following results: (1) every non-invertible square matrix over D is the product of four Hermitian matrices; (2) every matrix over D with determinant Ī admits a Hermitian factorization; (3) if ⁎ is of the second kind and either ΙFΙ ≥ n2 or n = 2, then an n × n matrix A over D is similar to its Hermitian transpose AΗ if and only if A is similar to AH via an invertible Hermitian matrix, if and only if A is the product of two Hermitian matrices; (4) a square matrix A over a field with involution is the product of four Hermitian matrices precisely when det (A) is symmetric. These results generalize and improve several known theorems, including those of Gatephan–Rodtes and Radjavi.
Bien, M.H., Ramezan-Nassab, M., Thi, N.A. and Zhou, Y. (2026) Linear Algebra and its Applications, 736, pp. 245–265.