# Does NDSolve automatically simplify the system of equations for Hermitian matices?

I am currently starting to solve numerically a system of differential equations like this:

$\dot{A}(t)=S(A)A(t)$

Here, the matrices are of dimension $d\times d$. $S(A)$ is some superoperator that acts on $A(t)$. What is important in my case is that $A$ is always hermitian, which means that I do in principle only need to solve the differential equations for e.g. the elements given by the upper triangular and get the rest from hermitian property. That would allow to solve only $\frac{1}{2}(d^2+d)$ equations instead of $d^2$ which might give some noticable speedup and also reduces memory consumption. I am wondering if NDSolve can automatically take account of this or if I have to feed it with the reduced system and put everything together manually.

I searched the doc but didn't see anything like this.

Details about reduction

To clarify what I mean with a manual reduction consider $\begin{pmatrix}\dot{a}_{11} & \dot{a}_{12}\\ \dot{a}_{21} & \dot{a}_{22}\end{pmatrix} = \begin{pmatrix} a_{11}- a_{21} & 1+ a_{12}\\ a_{22}-3 a_{21} & a_{11}\end{pmatrix}$

which is completely arbitrary. But since $a_{21}= a_{12}^*$ one can just replace the $a_{21}$ on the right handside, solve the differential equations given by the upper triangle and obtain the rest (in this example $a_{21}$ as the complex conjugate of the solution just found.

• I think the answer to your question is no. But I am not sure I understand you statement about doing it manually; how would you go about that? – user21 Oct 7 '15 at 5:59
• @user21 What I mean is that since $A=(a_{ij})$ is hermitian, it is valid to only look at the upper triangular matrix of the left and right handside of the differential equation. That is, replace all $a_{ji}$ with $a_{ij}^*$, solve for them and that's it. So this is some “manipulation“ I will then probably do byby hand in advance and feed the reduced system to NDSolve – Lukas Oct 7 '15 at 6:33
• Hm but is the aji not a∗ij to begin with - that's what makes it hermitian in the first place. But maybe I just need to get a coffee... – user21 Oct 7 '15 at 6:40
• @user21 Isn't that what I wrote? Except that the conjugate appears after the indices ;-) – Lukas Oct 7 '15 at 11:47
• Unfortunately, I am failing to understand that part of the question (the manual reduction) so I can not help here, but that does not seem terribly relevant as you seem to have an idea what needs to be done. If not, perhaps a more concrete example would help. – user21 Oct 7 '15 at 12:07