Suppose you multiply the values of $f$ between $a$ and $b$ at intervals of $\Delta x = (b-a)/n$ and then raise the product to the power $\Delta x$, and take the limit as $n\to\infty$. What you get would bear the same relation to products that integrals bear to sums, and there's a trivial reduction to ordinary integrals, in that what you get is $\displaystyle\exp\int_a^b\log f$, provided the addition and multiplication are reasonably like the ordinary ones on real numbers.
But what if the things getting multiplied are matrices, so the multiplication is non-commutative? I think I've seen it asserted somewhere ("somewhere" is a horrible word sometimes, isn't it?) that that is when such "product integrals" are not trivially reducible to more familiar things.
Last time I checked, Wikipedia's article titled product integral didn't go into matrix products, but only products of real numbers. If what I think I heard somewhere is right, then that makes it not all that good an article.
So:
- Is what I think I heard correct?
- If so, where's the literature on this?
- And what are such matrix-product integrals used for? Or, what is done with them?