At 04:18 PM 11/7/2002 -0800, Marc LeBrun wrote:
I forgot to mention that you can extend prodigals to matrices (with sufficient care--guess you have to use the analog of Lebesgue-Stieltjes integration or something).
This seems to be related to a way to handle continuous composition of rotations, for example. If you have two rotation matrices A and B, they don't in general commute, but lim n->oo ((A^1/n)(B^1/n))^n = lim n->oo ((B^1/n)(A^1/n))^n. [ One way to see this is that lim n->oo A^1/n = lim n->oo B^1/n = Identity ] How is this related to the product integral? If A and B are thought of as matrix-valued functions of a parameter t where, e.g., A(1/n) ^ n = A(1), B(1/n) ^ n = B(1), which makes easy to see geometric sense for constant rate rotations, then the product integral P (A(t)B(t))^dt seems to be the continuously combined rotation (at least according to one way of defining it), and seems to be equal to P (B(t)A(t))^dt. Taking the definite product integral between 0 and 1 would give the same value as the limits above. (I guess there's a multiplicative constant of integration in the indefinite product integral). Perhaps this extends to more general matrices, though it is easier to visualize for rotations (first, at least). Also, does this suggest that there is any more general ability to rearrange what would be ordinarily be non-commutative operators "under the P"? Shel