I have the following example, which is a proxy for the more complex problem I am trying to solve.(Apologies that the LaTeX is explicit, for some reason it trips the code formatting error on stackoverflow..)
$$ \dot{\Phi} = \Phi, \quad \Phi(0) = \mathbf{I} $$ where $\Phi$ is a square matrix, with identity initial condition. The analytical solution is an exponent for all the diagonal entries and zero for off-diagonal entries. Solving this in mathematica for $\Phi$ up to 10x10 behaves exactly as expected. However, for 11x11 and above, I get
DSolve::overdet: There are fewer dependent variables than equations, so the system is overdetermined The 10 by 10 example runs in a second or so. My guess is there is something interesting happening when we get to 100 equations, but I can't tell what it is. If anyone knows why this is, or knows how to debug it, would be greatly appreciated. The code is below.
t = Symbol["t"]; phiMatrix = Table[ToExpression["\[Phi]" <> ToString[i] <> ToString[j]][t], {i, 1, 15}, {j, 1, 15}]; nStates = 11; phiSubMatrix = phiMatrix[[1 ;; nStates, 1 ;; nStates]]; eqsList = {}; Do[AppendTo[eqsList, D[phiSubMatrix[[i, j]], t] == phiSubMatrix[[i, j]]], {i, 1, nStates}, {j, 1, nStates}]; icList = {}; Do[AppendTo[ icList, (phiSubMatrix[[i, j]] /. t -> 0) == KroneckerDelta[i, j]], {i, 1, nStates}, {j, 1, nStates}]; phiVars = Flatten[phiSubMatrix]; Dimensions[eqsList] Dimensions[icList] Dimensions[phiVars] solution = DSolve[Join[eqsList, icList], phiVars, t] 
NDSolvesupports that:DSolve[{ϕ'[t] == ϕ[t], ϕ[0] == IdentityMatrix[11]}, ϕ[t] ∈ Matrices[{11, 11}], t]$\endgroup$