论文标题
几何代数中特征多项式系数的基本公式
Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras
论文作者
论文摘要
在本文中，我们在（Clifford）几何代数$ {\ Mathcal {g}} _ {在$ n \ leq 6 $的情况下，我们提供了所有特征多项式系数的无基本公式，以及获得这些公式的一般形式的方法。该公式仅涉及几何产品，共轭的总和和操作的操作。使用计算机计算验证所有公式。我们在情况下提供了所有公式的分析证明，$ n = 4 $，以及一个情况下的公式之一$ n = 5 $。我们介绍了共轭和等级投影运营的一些新属性，并使用它们来获取本文的结果。在某些特殊情况下，我们还提出了特征多项式系数的公式。特别是，在任意$ n $的情况下，介绍了向量的公式（$ 1 $ $ 1 $）和基本元素，在$ n \ leq 5 $的情况下，介绍了转子的公式（旋转组的元素）。本文的结果可用于在计算机图形，计算机视觉，工程和物理学中的几何代数的不同应用中。特征多项式系数的无基本公式也可以用于符号计算中。
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.
