I am doing research in Quantum Field Theory and I ended up getting the following equation in the context of some gauge theory calculation:
$$\text{d}U=-U\cdot\Omega^\vee\cdot U,$$
where $U$ is an unknown $n\times n$ matrix (you can think about it as a change of basis matrix) and $\Omega^\vee$ is a known 1-form valued $n\times n$ matrix (can think of it as some connection matrix). I know that if $U$ was a vector (forgetting about the $U$ on the right most part of the equation) the solution would be a path-ordered exponential and that I know how to handle. But, with $U$ being a matrix, that seems to go beyond my current knowledge of differential equations.
I was wondering if the solution of such differential equation is known or well studied. I would really like some references if possible.
Thanks.
This is a matrix Riccati differential equation.
Let $V = U^{-1}$, then $dV = -U^{-1} dU U^{-1} = Q^{\vee}$. You can take it from there.