Question: Can we guarantee a solution to this system?
I have a system of compartments as shown in the Figure (for $N=3$).
The mass flow among the compartments can be described by
$$ \frac{d}{dt}x_{i} = \sum_{j}Z_{ji} \thinspace x_{j} - \sum_{j}Z_{ij} \thinspace x_{i} $$
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or, for the entire system ($\mathbf{x} = [x_1,x_2,x_3]$), $\frac{d}{dt}\mathbf{x} = {(\mathbf{x}^T Z)}^{T} - Z \mathbf{x}$
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Goal: We want to solve for $Z$, knowing the values of the two vectors (i) $\frac{d}{dt}\mathbf{x} = \mathbf{b}$, and (ii) $\mathbf{x}$.
Approach: If we define a vector $\mathbf{z} = \mathrm{vec}(Z)$, then we can factor the RHS of the above to a linear system like:
\begin{align} \frac{d}{dt}\mathbf{x} = \mathbf{b} &= A \thinspace \mathbf{z} \\ &\quad \textrm{(e.g. for $N=2$)}\\ &= \left[\begin{array}{cccc} \cdot & -x_1 & x_2 & \cdot \\ \cdot & x_1 & -x_2 & \cdot \end{array}\right] \left[\begin{array}{c} Z_{11} \\ Z_{12} \\ Z_{21} \\ Z_{22} \end{array}\right] \\ &\quad \textrm{(e.g. for $N=3$)}\\ &= \left[\begin{array}{ccccccccc} \cdot & -x_1 & -x_1 & x_2 & \cdot & \cdot & x_3 & \cdot & \cdot \\ \cdot & x_1 & \cdot & -x_2 & \cdot & -x_2 & \cdot & x_3 & \cdot \\ \cdot & \cdot & x_1 & \cdot & \cdot & x_2 & -x_3 & -x_3 & \cdot \end{array}\right] \left[\begin{array}{c} Z_{11} \\ Z_{12} \\ Z_{13} \\ Z_{21} \\ Z_{22} \\ Z_{23} \\ Z_{31} \\ Z_{32} \\ Z_{33} \end{array}\right] \end{align}
Then, we can solve for $\mathbf{z}$ using the usual methods. The diagonal elements will be undefined, but that's fine as they are irrelevant to the compartment flows (in my case).
Specific Questions:
- It seems $A$ is always Rank $N-1$; is this true? Why?
- For any choice of $\mathbf{b}$ and $\mathbf{x} \implies A$, can we guarantee a solution $\mathbf{z} \implies Z$?
- If not, can we derive constraints on $\mathbf{b}$ and / or $\mathbf{x}$ to guarantee a solution?
Thanks,
