Similarity reduction of matrix over a quaternion division ring

Let F be a field of characteristic ≠2, the quaternion division ring over F. This paper discusses the similarity of polynomials over HF. By using polynomials over HF, this paper proves that every square matrix over HF is similar to a uniquely generalized Jordan canonical form, and a quaternion matrix A is similar to a matrix over F if and only if A is similar to A∗ via a Hermite similarity transformation, or if and only if A is product of two quaternion Hermite matrices.