Matrix-product-integrals? Whereas the conventional "sum integral" is
$$
\lim_{\Delta x\to 0} \sum_i f(x_i)\,\Delta x,
$$
a "product integral" is
$$
\lim_{\Delta x\to 0} \prod_i f(x_i)^{\Delta x}.
$$
Now you're thinking: just take logarithms and it's a "sum integral", so why bother having this additional concept? Somewhere I heard this answer proposed: Because one does this with matrix multiplication rather than multiplication of numbers.
So:


*

*Is this actually done?  What's been published on it?

*What are the most interesting results about it?

*What's it's used for?

 A: The operation you are looking for is very useful in physics, probability, and geometry. In physics it goes under the name of "ordered exponential", but it is exactly the same: http://en.wikipedia.org/wiki/Ordered_exponential.
Its series expansion is called "Dyson series", and it approximates the same thing: http://en.wikipedia.org/wiki/Dyson_series
In differential geometry, it gives the parallel transport map from a connection (the parallel transport map is the "line product integral" of the connection form).
All the uses of such an operation can be traced back to the solution of the following equation:
$$
\frac{dX}{dt} = A(t)X(t),
$$
where $X$ is either a vector or a (group) matrix, and $A$ is a (Lie algebra) matrix.
Schrödinger's equation in quantum physics has this form, the parallel transport equation in differential geometry has this form (vanishing covariant derivative), and many more. The solution is:
$$
X(t) = \prod_0^t \exp \big( A(t')\,dt' \big)\,X(0).
$$
A very simple example is the following. If we have a rotation in the plane, we can define the angular velocity as:
$$
\omega := \frac{d\theta}{dt}.
$$
Therefore:
$$
\theta = \int_0^t \omega (t')\,dt' +\theta(0).
$$
If the rotation takes place in three dimension, this picture breaks down, because there is no "number" parametrizing the rotation angle.
We have a rotation matrix, $R\in SO(3)$, and its derivative is an element of the Lie algebra $\Omega \in so(3)$.
We have (can you see why?):
$$
R(t) = \prod_0^t \exp \big( \Omega(t')\,dt' \big).
$$
In the case of rotations in the plane everything was simple because $SO(2)$ is commutative, so the product integral, there, is exactly the exponential of the integral (just like with numbers).
Intuitively: if instead of "continuously summing" you need to "continuously compose group operations", you need a product integral. You can use an ordinary integral (and exponentiation) if and only if the group is commutative.
