标题： 无基公式在几何代数中特征多项式系数的表达式 显示英文标题

标题： Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

摘要： 在本文中，我们讨论向量空间维度为$n=p+q$的 (Clifford) 几何代数${\mathcal {G}}_{p,q}$中的特征多项式。我们在$n\leq 6$的情况下给出了所有特征多项式系数的无基公式，并提供了一种获得这些公式的通用形式的方法。这些公式仅涉及几何积、求和以及共轭运算。所有公式均通过计算机计算进行了验证。我们在$n=4$的情况下给出了所有公式的分析证明，并在$n=5$的情况下给出了其中一个公式的分析证明。我们提出了一些共轭运算和阶投影运算的新性质，并利用这些性质得到了本文的结果。我们还给出了一些特殊情况下的特征多项式系数公式。 特别是，给出了向量（grade$1$的元素）和基元素的公式，在任意$n$的情况下，旋转向量（自旋群的元素）的公式在$n\leq 5$的情况下给出。 本文的结果可以用于几何代数在计算机图形学、计算机视觉、工程和物理中的不同应用。 给出的特征多项式系数的无基公式也可以用于符号计算。 显示英文摘要

摘要： In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras ${\mathcal {G}}_{p,q}$ of vector space of dimension $n=p+q$. We present basis-free formulas for all characteristic polynomial coefficients in the cases $n\leq 6$, alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $n=4$, and one of the formulas in the case $n=5$. We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade $1$) and basis elements are presented in the case of arbitrary $n$, the formulas for rotors (elements of spin groups) are presented in the cases $n\leq 5$. The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.