TY  - CONF
A1  - Gonzalez Calvet, Ramon
A2  - Gürlebeck, Klaus
A2  - Lahmer, Tom
A2  - Werner, Frank
T1  - NEW FOUNDATIONS FOR  GEOMETRIC ALGEBRA
T2  - Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 04 - 06 2012, Bauhaus-University Weimar
N2  - New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea that Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n×n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multilinear quantities.
KW  - Angewandte Informatik
KW  - Angewandte Mathematik
KW  - Computerunterstütztes Verfahren
Y1  - 2012
UR  - https://e-pub.uni-weimar.de/opus4/frontdoor/index/index/docId/2764
UR  - https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20170314-27644
UR  - http://euklid.bauing.uni-weimar.de/ikm2012
SN  - 1611-4086
ER  -