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67) If any autovector can be written this way, we say that {e1 , e2 , . . , en } , is an autobasis, and that {v1 , . . , vn } , are the autocomponents of v on the autobasis {ei } . 68) v v v + ... , j k where only the terms up to order three have been written. Here, a is some fixed vector and b, c, . . 2). We shall see later how this series relates to a well known series arising in the study of Lie groups, called the BCH series. Remember that the operation ⊕ is not assumed to be associative.

32 The autovector structure introduced below will not cover the whole GL(n) manifold but only the part of GL+ (n) that is connected to the origin through autoparallel paths (that, in fact, are going to also be geodesic paths). For this reason, some of the geometric properties mentioned below are demonstrated only for a finite neighborhood of the origin. But as Lie group manifolds are homogeneous manifolds (any point is identical to any other point), the local properties are valid around any point of the manifold.

The two basic matrix representations of this group are the following. 9 Let SO(3) be the set of all orthogonal 3×3 real matrices with positive (unit) determinant. This is a (multiplicative) group with, as group operation, the matrix product R2 R1 . It is well known that this group is isomorphic to the group of geometrical 3D rotations. A geometrical rotation is then represented by an orthogonal matrix, and the composition of rotations is represented by the product of orthogonal matrices. 15 for details).

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Advanced Vector Analysis with Application to Mathematical Physics by C.E. Weatherburn

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