Is the solution to this 1st order matrix ODE uniquely determined?

Question

I have a matrix function $e(t)$ (i.e. for each $t$, $e(t)$ is a matrix) and the ODE $$ e'(t)=e(t)^{-T}g(t) $$ where $-T$ denotes the inverse transpose, and $g(t)$ is some fixed matrix function. My question is if given the initial condition $e(0)$, whether $e(t)$ is determined uniquely? The answer is probably yes -- usually one could go so far as to integrate each side using the initial condition and then iterate the expression to obtain an 'explicit' expression for $e(t)$ (the path ordered exponential for example), but I am abit put-off by the appearance of the inverse $e^{-1}$, hence my slight apprehension.

Answer 1

The usual Picard-Lindelöf existence and uniqueness theorem for ODE systems applies, assuming of course $e(0)$ is invertible, so $e(t)$ is determined uniquely on some interval around $0$. However, the solution may "blow up" or become singular at some finite $t$.