From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: orthog. matrix parameterization Date: 9 Aug 1996 04:33:54 GMT In article <320A18BE.454D@econ.vu.nl>, Andre Lucas wrote: >Hi, > >I'm looking for a solution to the following problem. Let A be an nxn >orthogonal matrix. I know that A has n(n-1)/2 free parameters, e.g., >for n=1 we get A=1, and for n=2 we get >A = (sin(f) cos(f)) for some f. > (cos(f) -sin(f)) >Is there a parameterization for orthogonal matrices that works for >general n and that is continuous (and preferably differentiable)? I came >up myself with a polar type decomposition, but this one isn't continuous >if some of the columns of A coincide with columns of the identity matrix. How about the exponential map from the corresponding Lie algebra, in this case the set o_n(R) of skew-symmetric n x n matrices? This is precisely the map above, when n=1 or 2. (I take it you only want a parameterization of a neighborhood of the identity since in both cases above you have excluded the orthogonal matrices of determinant -1). So for example for n=3 we have the following expression giving 3 x 3 orthogonal matrices: ( 0 -a -b ) exp( ( a 0 -c ) ) = I + A + A^2/2 + A^3/3! + ... (A = the 3x3 matrix) ( b c 0 ) Since these algebras are nonabelian for n>2 the nine coordinate functions of a b and c are not expressible as combinations of one-variable functions of a, b, and c. They are differentiable, and probably even have been named, but no one asked me what to name them so I don't know what was decided :-). By the way there are a lot of topologcal obstructions to getting 'nice' parameterizations in general. For some n we can get a little bit nicer. For example, for n=4 we may do the following. Use the standard spherical coordinates to give a parameterization of points p in the 3-sphere S^3 (in R^4). Viewing R^4 as the set of quaternions, we make a matrix whose columns are 1*p, i*p, j*p, and k*p. These four are orthogonal vectors of length one, and so the matrix with them as columns is an orthogonal matrix. Naturally the coordinates of the four columns are signed permutations of those in the first column, so we have a parameterization of a family of matrices M(p) which "extends" the parameterization of the points p of the sphere. Then we may parameterize the whole of O_4 as the set of products M(p) * Q, where Q is the set of orthogonal matrices with (1, 0, 0, ..., 0) as their first row, with the lower 3x3 block being a parameterization of O_3 as in the previous paragraph. Thus we don't need functions of 6 variables to parameterize O_4, just the 3-variable ones and the 1-variable trig functions used for spherical coordinates in R^4. (The experts will recognize that I am using here the general fibration SO_{n-1} -> SO_n -> S^{n-1} together with a splitting of this fibration in a rare case in which S^{n-1} has the maximum number n-1 of linearly independent [tangential] vector fields. I don't know how to quantify this amount of success -- some relation between the minimal number of variables in the functions, and the maximal number of linearly independent vector fields over the sphere. 'Minimal number of variables' is an awkward concept, as noted in a recent thread about the Hilbert problem addressing that issue.) dave ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Newsgroups: sci.math Subject: Re: orthog. matrix parameterization Date: 10 Aug 1996 11:03:05 -0400 In article <320C1B2A.206F@irelands-web.ie>, Marilyn Hesse wrote: :The question was: Is there a parmeterization of the space of n by n :orthogonal matrices? : :What do you mean by a parameterization? A map from the n * (n-1) / 2 :dimensional real space onto the space of matrices that is continuous? :That probably exists, but will not be one-to-one and may be forced :to have a singular derivative at some point. A continuous map of the :space of matrices onto the real vector space is impossible, since the :image of a compact space is compact. If it is onto a subset of the :parameter space, it is locally not a homeomorphism at the edge of its :image. If you settle for a reasonably large (open, although not always dense) subset of the set of orthogonal matrices, there is a rational parametrization due to Cayley: Suppose U is an orthogonal matrix such that (-1) is not its eigenvalue. Then ("/" means multiplication on the right by the inverse of the second operand) U |-> H := (U-I)/(U+I), H |-> U := (I+H)/(I-H) is a bijection onto the set of all skew-symmetric matrices H, which is parametrized by the entries of its upper triangle. This "local chart at identity" can be transferred to a neighborhood of any orthogonal matrix V0 by parametrizing U := V/V0 where V is so close to V0 that I + V/V0 is invertible. The domains of these charts cover the whole set of orthogonal matrices (a finite subcover can be found, of course; how large is it?) Hope it helps, ZVK (Slavek).