AI 中文总结
研究对偶正交三幂等矩阵，通过探讨其基本代数性质，利用特定矩阵平均值建立表征，丰富了对偶广义矩阵类理论，为相关对偶四元数矩阵研究提供新视角。
AI 中文摘要
本文研究由\(\hat{A}^3=\hat{A} = \hat{A}^*\)定义的对偶正交三幂等矩阵，探讨其基本代数性质。此外，利用涉及\(\hat{A}\)、\(\hat{A}_e\)、\(\hat{A}^*\)、\(\hat{A}\sp{\scriptscriptstyle N}\)以及诸如\(\hat{A}\hat{A}^*\)和\(\hat{A}^*\hat{A}\)等乘积的整数幂的矩阵平均值，建立了这类矩阵的几种表征。这些结果丰富了对偶广义矩阵类理论，并揭示了相关对偶四元数矩阵的新视角。
英文摘要
In this paper, we study dual orthogonal tripotent matrices, defined by $\hat{A}^3=\hat{A} = \hat{A}^*$, and examine their fundamental algebraic properties. Additionally, we establish several characterizations of this class of matrices using matrix averages involving $\hat{A}$, $\hat{A}_e$, $\hat{A}^*$, $\hat{A}\sp{\scriptscriptstyle N}$, as well as integer powers of products such as $\hat{A}\hat{A}^*$ and $\hat{A}^*\hat{A}$. These results enrich the theory of dual generalized matrix classes and reveal new perspectives on related dual quaternion matrices.
