I am modeling the effect of neural activity on synaptic strength. My question, though, is mathematical.
I have the following differential equation:
$ \tau_{W} \frac{d\mathbf{W}}{dt}=\mathbf{KWQ}-\alpha\mathbf{R_{\infty}}\mathbf{W} \quad\mathbf{Q}=\left\langle \vec{u}\vec{u}\right\rangle ,\,\mathbf{R_{\infty}}=\left\langle \vec{v_{\infty}}\vec{v_{\infty}}\right\rangle $
$\tau_{W}$ and $\alpha$ are constants. $\mathbf{W}$ is $n \times m$. $\mathbf{R_{\infty}} $ is $n \times n$. $\mathbf{Q}$ is $m \times m$. $\mathbf{K}$ is $ n \times n$.
If $\mathbf{Q}$ and $\mathbf{R_{\infty}}$ come from the outer products of vectors, how do I know that they are invertible?
How would I find the value for $\mathbf{W}$ for which the derivative vanishes? Especially if neither $\mathbf{Q}$ nor $\mathbf{R_{\infty}}$ are invertible.
- How would I find $\mathbf{X}$ such that $ \mathbf{X}\mathbf{W} = \mathbf{Q}$ if neither $\mathbf{Q}$ nor $\mathbf{W}$ are invertible?