The $φ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization

Let $φ$ be a quaternion of modulus $1$. In this article, we study some topics related to $φ$-conjugation for quaternionic matrices, including $φ$-Hermitian matrices, $φ$-conjugate normal matrices, unitary $φ$-congruence and $φ$-HSH decomposition (decomposition of a $φ$-Hermitian matrix and a skew $φ$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $φ$-Hermitian matrices for all unit quaternion $φ$. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.