Canonical forms of two by two by two matrices
The aim of this paper is to determine the canonical forms of two by two by two matrices. Firstly, an invariant \(\Gamma\) of two by two by two matrices is introduced. It is shown by a Gaussian elimination argument that any invariant of two by two by two matrices has the form \(s\cdot\Gamma^k\) where \(s\) is a constant and \(k\) is a nonnegative integer. Then 6 covariants are presented in order to separate the canonical forms. After this the author presents 7 different canonical forms for the two by two by two matrices. Among that, \(pqr+stu\) is a generic canonical form. Hence it is easy to prove certain relations between these covariants. In fact, the binary cubic may be viewed as symmetric two by two by two matrices. Thus there is a natural correspondence between the canonical forms and covariants of two by two by two matrices and binary cubics. Similarly, two by two by two matrices are also related to skew-symmetric tensors of step 3 in the exterior algebra Ext\((U)\), where \(\dim (U)=6\).
scientific article; zbMATH DE number 1270880