If the differential equation governing the time dependent matrix $M(t)$ is
$\frac{dM(t)}{dt}=A.M(t).B$
or
$\frac{dM(t)}{dt}=A.M(t)+M(t).A$
where $A$ and $B$ are constant matrices, what is the differential equation governing $M^{1/2}(t)$ in both cases?
I'm assuming the equation somehow guarantees that $M(t)$ is positive definite; otherwise it's not clear what $M^{1/2}(t)$ is. Let $N(t)=M^{1/2}(t)$. Since $M=N^2$, the product rule yields $\dot M=N\dot N+\dot NN$. Hence, $N$ satisfies the nonlinear equation $$N\dot N+\dot NN = AN^2B\qquad \text{or} \qquad N\dot N+\dot NN = AN^2+N^2B$$ Admittedly, neither is a nice thing to look at.
If $A$ and $B$ happen to commute with $M(0)$, then they commute with $M(t)$ for all $t$ (the trajectory lies in the commutant of the algebra generated by $A$ and $B$). In this case, $N$ and $\dot N$ commutes with $A$ and $B$ as well, and the above equations simplify to linear equations $$2\dot N = ANB\qquad \text{or} \qquad 2\dot N= AN+NB$$ which are linear. This is what happens in the scalar case: if $\dot x=cx$ and $x$ is positive, then $u=\sqrt{x}$ satisfies $2\dot u= cu$.