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

Abstract

Let $\varphi$ be a quaternion of modulus $1$. In this article, we study some topics related to $\varphi$-conjugation for quaternionic matrices, including $\varphi$-Hermitian matrices, $\varphi$-conjugate normal matrices, unitary $\varphi$-congruence and $\varphi$-HSH decomposition (decomposition of a $\varphi$-Hermitian matrix and a skew $\varphi$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $\varphi$-Hermitian matrices for all unit quaternion $\varphi$. 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''.

23 pages