Prove unitary evolution for an ODE

Question

Consider a nonlinear ODE of square matrix $A(t)=(a_1(t),\cdots,a_n(t))^T$

$$\mathrm{i}\,\dot A(t) = A(t) M(t)$$

with $$M(t) = A^\dagger(t) H(t) A(t)$$

where $H(t)$ and hence $M(t)$ are Hermitian matrices. My question is whether the time evolution of $A(t)$ is unitary. Here by unitary I mean the inner product of the vectors $$\langle a_i(t),a_j(t)\rangle$$ is invariant. The case for a constant Hermitian $M$ is straightforward, but how about this more general case?

Answer 1

Consider the function $G(t)=A(t)A(t)^†$. Then its derivative is $$ \dot G=\dot AA^†+A\dot A^†=(-iAM)A^†+A(-iAM)^†=-iAMA^†+iAM^†A^†=0 $$ So if $G(0)=I$, then this is also true for all $t$.