ICCA2020:P000146

On determinant, other characteristic polynomial coefficients, and inverses in Clifford algebras

*Dmitry Shirokov (National Research University Higher School of Economics, Moscow, Russia; Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia)

We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverses in real Clifford algebras over vector space of arbitrary dimension $n$. The formulas involve the operations of multiplication, summation, and operations of conjugation without using the corresponding matrix representations. We use numerical methods of the matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials), the method of operations of conjugation of special type, and the method of quaternion typification in Clifford algebras. The basis-free formulas for the inverses give us the 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. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverses in real Clifford algebras over vector space of arbitrary dimension $n$. The formulas involve the operations of multiplication, summation, and operations of conjugation without using the corresponding matrix representations. We use numerical methods of the matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials), the method of operations of conjugation of special type, and the method of quaternion typification in Clifford algebras. The basis-free formulas for the inverses give us the 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.

This work is supported by the grant of the President of the Russian Federation (project MK-404.2020.1), the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2020 - 2021 (grant 20-01-003), and by the Russian Academic Excellence Project ``5-100''.

We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverses in real Clifford algebras over vector space of arbitrary dimension $n$. The formulas involve the operations of multiplication, summation, and operations of conjugation without using the corresponding matrix representations. We use numerical methods of the matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials), the method of operations of conjugation of special type, and the method of quaternion typification in Clifford algebras. The basis-free formulas for the inverses give us the 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. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverses in real Clifford algebras over vector space of arbitrary dimension $n$. The formulas involve the operations of multiplication, summation, and operations of conjugation without using the corresponding matrix representations. We use numerical methods of the matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials), the method of operations of conjugation of special type, and the method of quaternion typification in Clifford algebras. The basis-free formulas for the inverses give us the 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. This work is supported by the grant of the President of the Russian Federation (project MK-404.2020.1), the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2020 - 2021 (grant 20-01-003), and by the Russian Academic Excellence Project ``5-100''.

The 12th International Conference on Clifford Algebras and Their Applications in Mathematical Physics

On determinant, other characteristic polynomial coefficients, and inverses in Clifford algebras