After solving the dynamics of a system, I arrived at an apparently simple system of differential equations of the form:
$$ \tau \frac{d}{dt}\vec{w}=Q\vec{w}+\vec{c} $$
where $ \vec{c}\neq\vec{c}(t) $ and $Q$ is a NOT positive definite matrix.
Numerical simulations show that this equation is indeed correct for the system I am modelling.
If $Q$ would be positive definite, the solution would be trivial, the dynamics of $\vec{w}$ would be a linear sum of the eigenvectors of $Q$, with separate odes for the coefficients. I would like to know if there is a known similar approach for the case in which $Q$ is not positive definite. I have been trying (for days) to solve it by doing SVD for $Q$:
$$ Q=USV^T $$
and then trying to arrive at a system of the form:
$$ \tau \frac{d}{dt}\vec{\tilde{w}}=S\vec{\tilde{w}}+\vec{\tilde{c}} $$
where $\tilde{w}$ is an appropriate transformation of $w$ that satisfies this equation ($S$ is a diagonal matrix of the singular values of $Q$). Of course, I haven't been able to find such a transformation.
Ideally, I would like to know if there is an established approach for this problem (maybe I am missing something big...), or at least if it is known to be a problem without a general solution. You would be my saviour and I will be eternally grateful for your aid!
Assuming $Q$ is invertible, the general solution is $$ \vec{w} = - Q^{-1} \vec{c} + \exp(Q t/\tau) \vec{v}$$ for arbitrary vectors $\vec{v}$. There are many ways of calculating the matrix exponential.