发表机构
Changchun University of Chinese Medicine; Jilin University（长春中医药大学; 吉林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文受复矩阵酉等价于实矩阵的启发，将其推广至四元数，提出$\boldsymbol{i}$-共轭概念，给出四元数矩阵酉等价于复矩阵的充要条件，并研究相关性质及新型极分解。
AI 中文摘要
受复数$n\times n$矩阵$A$酉等价于实矩阵这一结果的启发，本文将这一结论推广到四元数除环上，给出了四元数$n\times n$矩阵$A$酉等价于复矩阵的一个充分必要条件。为更清晰地陈述该结论，我们提出了称为$\boldsymbol{i}$-共轭的概念。进一步地，我们研究了与$\boldsymbol{i}$-共轭相关的概念及其性质，例如$M_{n}(\mathbb{H})$中的酉$\boldsymbol{i}$-合同、$\boldsymbol{i}$-共轭正规性和$\boldsymbol{i}$-埃尔米特性，它们分别是$M_{n}(\mathbb{C})$中矩阵的常规酉合同、共轭正规性和埃尔米特性的推广。最后，我们给出了四元数矩阵的一种新型极分解。
英文摘要
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.
Comments23 pages
