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.

在本文中，我们解决了在任意维数的 Clifford 代数中计算逆的问题。我们为任意维数n 的向量空间上的实数 Clifford 代数（或几何代数）中的行列式、其他特征多项式系数、调节和逆提供不同类型（显式和递归）的无基公式. 公式只涉及乘法、求和运算和共轭运算，没有明确使用矩阵表示。我们使用 Clifford 代数的方法（包括作者在前几篇论文中提出的四元数典型化方法和本文提出的特殊类型共轭运算方法）和矩阵理论数值方法的推广（基于 Faddeev-LeVerrier 算法关于 Cayley-Hamilton 定理；使用 Bell 多项式计算特征多项式系数的方法）到本文中的 Clifford 代数的情况。我们提出了特殊类型共轭运算的构造，并研究了这些运算与 Clifford 代数固定子空间上的投影运算之间的关系。我们在 Clifford 代数中的行列式、其他特征多项式系数、调节和逆的公式的分析证明中使用此构造。逆的无基公式为我们提供了线性代数方程的无基解，广泛应用于计算机科学、图像和信号处理、物理、工程、控制理论等领域。 本文的结果可用于符号计算。