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

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.VeselĂ˝

### 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)

Zbl 0472.35075