Methods for inversion of matrices of order 2 n. (Russian) Zbl 0639.65018

Pshenichnikov, S. B.; Yamaleev, R. M.

Zh. Vychisl. Mat. Mat. Fiz. 27, No. 8, 1244-1248 (1987).

A method for inversion of matrices of order 2 n is presented exploiting the well-known fact that the algebra of all real matrices of order 2 n is isomorphic to the algebra of alternions \({\mathcal A}(2^{2n})\) of dimension \(2^{2n}\). The authors recall some properties of \({\mathcal A}(2^{2n})\) and give an inductive construction of the matrices which represent the 2n generating elements of the ring \({\mathcal A}(2^{2n})\), and which constitute the base set \(\{R_ K\}^{2n}_{k=1}\) of the algebra \({\mathcal A}(2^{2n})\). Two algorithms for inversion of a matrix \((1)\quad A_ n=\sum^{2^{2n}}_{k=1}a_ kR_ k\) of order 2 n are described. One is based on a stepwise elimination of basic elements \(R_ k\) from the equation \((\sum^{2^{2n}}_{k=1}a_ kR_ k)A_ n^{-1}=E\), the other is a recursive procedure in the k-th step of which \((k=n,n- 1,...,1)\) the problem of inverting a matrix \(A_ k\) of order 2 k is converted to the problem of inverting two matrices of order \(2^{k-1}\). The operation count is \(O(2^{4n})\), which is more than \(O(2^{3n})\) of the Gaussian elimination. The presented method may be advantageous in situations where the representation (1) is inherent [cf. e.g. A. A. Bogush, Introduction to the field theory of elementary particles (1981; Zbl 0472.35075)].

Reviewer: V.Veselý

MSC:

65F05	Direct numerical methods for linear systems and matrix inversion
15A09	Theory of matrix inversion and generalized inverses
15A66	Clifford algebras, spinors
81T05	Axiomatic quantum field theory; operator algebras
15A90	Applications of matrix theory to physics (MSC2000)

Keywords:

matrix inversion; Clifford algebra; algebra of alternions; stepwise elimination; Gaussian elimination

Citations:

Zbl 0472.35075