[ Partly in reply to WDS and DA -- ] I should have emphasised that t takes independent values at every occurrence of any R_i ; this is necessary to ensure that the freedom of their composition is no less than the group dimension n_C_2 . And I might clarify that we seek an analogy to an orthogonal basis of a vector space, but in a non-commutative context. [ It is a consequence of Chasles' theorem that the maximum number of commutative rotations equals floor(n/2) , far fewer than n_C_2 . ] So the word / composition of R_i is required to have bounded length, of order n_C_2 . Iterative processes such as Jacobi or Givens reduction (which anyway apply to symmetric rather than orthogonal matrices) seem unserviceable in this context. Question (B) asks for the maximum (over elements of the group) of the minimum (over all words representing the element) of the word length: at least n_C_2 by the dimension argument. For example, here's an approach I explored over the weekend. Forget about restricting the generators to R_i for now; given some numerical rotation (matrix) Y , form the formal product X of all n_C_2 Givens rotations R_ij(t_ij) , then solve equation X(t) = Y numerically for the t_ij . [ Though perfectly practicable for (say) n = 4 , it doesn't actually work, not in this naïve formulation anyway! ] WFL On 5/15/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The following questions arose as a result of an enquiry raised on the list Geometric_Algebra <geometric_algebra@googlegroups.com> ---
SO(n) denotes the proper isometries of an (n-1)-sphere, embedded in standard fashion in a Cartesian coordinate basis (x_1,...,x_n); R_i(t) denotes rotation through angle t about axis the coline meet of coordinate hyperplanes x_(i-1), x_i . [ Ambiguity in rotation sense need not concern us just now. ]
Question (A) : Does the set G = { R_2, ..., R_n } generate SO(n) (irredundantly)?
Question (B) : Given an arbitrary isometry, is its minimum length as a word over G necessarily at most n_C_2 = n(n-1)/2 (sharply)?
Question (C) : Is there a practical algorithm expressing an isometry, given in the form (say) of a numerical matrix, as a word over G ?
Fred Lunnon