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? 
>>

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