# Uniqueness of a constrained system of linear equations

I would like to determine whether there exists a solution (and if so, check uniqueness) to the following system of linear equations (with respect to $$\eta = (\eta_1,...,\eta_J))$$:

\begin{aligned} \sum_{j=1}^J (\psi_i -y_j)a_{ij}\eta_j &= 0, \hspace{1em} \forall i=1,...,I, \\ \sum_{j=1}^J \eta_j &= 1, \\ \eta_j &\geq 0, \hspace{1em} \forall j=1,...,J \end{aligned},

where there are the following constraints:

• $$a_{ij} \in [0,1] \hspace{1em} \forall i,j$$
• if we rearrange y's in an increasing order $$y_1 < y_2 < ... < y_J$$, then the following inequalities hold $$y_1 \leq \psi_i \leq y_J, \hspace{1em} \forall i=1,...,I$$.

Is there a way to determine whether solution exists in general case for any $$I$$ and $$J$$? And if there are solutions - how to determine uniqueness?

I solved the most simple case for $$I=J=2$$. I "deleted" the $$I$$th equation and solved the following system:

\begin{aligned}(\psi_1 - y_1)a_{11}\eta_1 + (\psi_1-y_2)a_{12}\eta_2 &= 0, \\ \eta_1+\eta_2 &= 1, \\ \eta_1 &\geq 0, \\ \eta_2 &\geq 0. \end{aligned}

Assuming without loss of geenrality that $$y_1 < y_2$$ we get the following result: \begin{aligned}\eta_1 &= \frac{a_{12}(y_2-\psi_1)}{a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1)}, \\ \eta_2 &= \frac{a_{11}(\psi_1-y_1)}{a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1)}. \end{aligned}

$$\eta$$'s must be nonnegative, so this implies that either of the following two conditions must hold:

• $$a_{12}(y_2-\psi_1) \geq 0$$, $$a_{11}(\psi_1-y_1) \geq 0$$, and $$a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1) \geq 0 ;$$

• $$a_{12}(y_2-\psi_1) \leq 0$$, $$a_{11}(\psi_1-y_1) \leq 0$$, and $$a_{11}(\psi_1-y_1) + a_{12}(y_2-\psi_1) \leq 0 ;$$

which due to the constraints on $$a_{ij}$$'s lead to the equivalent conditions:

• $$y_1 \leq \psi_1 \leq y_2$$ and $$a_{11}(\psi_1 - y_1) + a_{12}(y_2-\psi_1) \geq 0$$;

• $$y_2 \leq \psi_1 \leq y_1$$ and $$a_{11}(\psi_1 - y_1) + a_{12}(y_2-\psi_1) \leq 0$$.

And now it is easily seen that the first case occurs when $$y_1 \leq \psi_1 \leq y_2$$ and the second case cannot occur, since $$y_2 > y_1$$. If this solution does not satisfy the "deleted" equation, then there is no solution. But if it does, then the solution is unique.

• The dash in “the following inequalities hold - $y_1≤ψ_i≤y_J,\quad∀i=1,\ldots,I$” is somewhat misleading. Jul 18, 2021 at 14:17
• @Saad Right, I edited the question, it was not a minus sign. Thanks. Jul 18, 2021 at 18:18