# A generalization of Liouville formula for the determinant of a system of ODE?

Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE): $\frac{d}{dt}y= A(t) y$, where $A(t)$ is a $t$-dependent $n\times n$ complex matrix.

Then, the Liouville formula provides a very simple relation between the determinant of $\Phi(t)$ at different $t$: $\det(\Phi(t)) = e^{\int_0^t Tr A(s) ds} \det(\Phi(0))$.

Now, consider the alternative system of ODE:

$\frac{d}{dt}y= A(t) y + B(t) \bar{y}$,

where $\bar{x}$ represents the complex conjugate of $x$. (Actually, I am interested only in the case where $A(t)$ is diagonal and $B(t)$ is skew-symmetric, if that helps.)

Does it exist any generalization of the Liouville formula applicable to this case? or any other expression for $\det(\Phi(t))$, which is simpler (to compute numerically) than finding the explicit solutions for all the starting conditions in $\Phi(0)$ and then computing the determinant explicitly?

The kind of manipulations used by the classical proof of Liouville formula do not seem very helpful to me, in this slightly modified case, unfortunately.

The solution of this problem would be very beneficial to the method described in arXiv:1205.3996, and to its very relevant physical applications.

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You're right. My answer was an error. Even if we split the system into real and imaginary parts, we obtain a $2n\times$-system of which we know only $n$ solutions, given by the real/imaginary parts of the columns of $\Phi$. This is insufficient to get any information. The point is that because your system is not $\mathbb C$-linear, but only $\mathbb R$-linear, you need a list of $2n$ solutions (instead of $n$), to write something non-trivial. Think about the scalar case $n=1$ : you can't give $y(t)$ in terms of $y(0)$ in close form. – Denis Serre Jul 3 '12 at 16:46
Does $B$ "skew-symmetric" mean that $B^T=-B$, or is it conjugate transpose $B^* = -B$? And is $A$ real? – Bob Terrell Oct 17 '12 at 22:15