$\boldsymbol{i}$-conjugate for quaternionic matrices and related properties

Abstract

Motivated by the result that a complex $n\times n$ matrix $A$ being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion $n\times n$ matrix $A$ is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call $\boldsymbol{i}$-conjugate. Furthermore, we study the concepts related to $\boldsymbol{i}$-conjugate and their properties, such as unitary $\boldsymbol{i}$-congruence, $\boldsymbol{i}$-conjugate normality and $\boldsymbol{i}$-Hermicity in $M_{n}(\mathbb{H})$ as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in $M_{n}(\mathbb{C})$. Finally, we present a new type of polar decomoposition of quaternion matrices.