Shel Kaphan (sjk@kaphan.org) asks: << [Extending product integrals to matrices] 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?

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It is related to something even simpler than the product integral. If Q is a matrix in SO(n,R), we call it "eligible" if Q does not carry any vector into its negative. Then there is a unique "smallest" skew-symmetric matrix Log(Q) such that Q = exp(Log(Q)) where exp is given by the usual series formula. Then for eligible A and B, we have (*) lim n->oo ((A^1/n)(B^1/n))^n = exp( (Log(A) + Log(B)) / 2 ). (With the understanding that for Q in SO(n,R), for rotation on any invariant plane P of Q, we always take the rotation angle theta to lie in [0,pi), and Q^(1/n) restricted to P is always taken to be rotation by theta / n.) The equation (*) works for other compact matrix groups G but with an appropriate definition of "eligible" that, as with G = SO(n,R), includes all but measure 0 of of G. --Dan