任意维Clifford代数的行列式、特征多项式系数及逆的计算
On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension
摘要：
本文解决了任意维Clifford代数的逆的计算问题。本文给出了任意维向量空间$n$上实Clifford代数（或几何代数）的行列式、其他特征多项式系数、伴随和逆的不同类型（显式和递归）的无基公式.这些公式只涉及乘法、求和和共轭运算，而没有明确使用矩阵表示。我们使用Clifford代数的方法（包括作者在以往文献中提出的四元数类型化方法和本文提出的特殊类型共轭运算方法）和矩阵论数值方法的推广（基于Cayley-Hamilton定理的Faddeev-LeVerrier算法;利用Bell多项式计算特征多项式系数的方法）推广到Clifford代数的情形。给出了一类特殊类型共轭运算的构造，并研究了它们与Clifford代数不动点子空间上的投影运算之间的关系。我们使用这种结构中的行列式，其他特征多项式系数的公式的分析证明，辅，逆Clifford代数。无基求逆公式给出了线性代数方程组的无基解，广泛应用于计算机科学、图像与信号处理、物理学、工程学、控制理论等领域。
英文摘要：
In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification proposed by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and generalizations of numerical methods of matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials) to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in Clifford algebras. The basis-free formulas for the inverse give us basis-free solutions to linear algebraic equations, which are widely used in computer science, image and signal processing, physics, engineering, control theory, etc. The results of this paper can be used in symbolic computation.
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