How to write ODE for matrices?

Question

This problem must have been encountered by many people but I could not find a neat solution googling.

Many problems in mathematical physics can be written as a matrix ordinary differential equation (ODE), i.e. as $$ \frac{d A}{dt} = f(A) $$ with initial condition $$ A(0) = A_0 $$ where $A$ (and $f(A)$) is an $n\times n$ matrix. For simplicity I consider the case where the function $f$ is not explicitly time dependent, as in the above equation. In many circumstances it is easy to produce the function $f$ with matrix operations. For example for the Schroedinger equation one has $$f(A) = -i[H,A]$$ (square brackets represent the commutator).

Here is the question: what is a neat way to pass the ODE above to NDSolve ? In my situations the matrix $A$ (and $f(A)$) are hermitian, so it suffices to send to NDSolve the $n$ real diagonal terms and the $n(n-1)/2$ complex terms in the upper diagonal. Essentially I am trying to find an elegant way to extract equations out of a matrix for general $n$.

Answer 1

Providing an example is always useful. Here is a made up example:

SeedRandom[1];
ic = RandomReal[1, {3,3}];
H = (#+ConjugateTranspose[#])&@RandomComplex[{-1-I, 1+I},{3,3}];

f[A_, B_] := -I (A.B - B.A)

Then:

sol = NDSolveValue[{A'[t] == f[A[t], H], A[0] == ic}, A, {t, 0, 1}];

Visualization:

ReImPlot[sol[t], {t, 0, 1}]