Given a system of normal equations $A^TAx=A^Tb$ where $A^TA$ is $n\times n$,
what I call the $i$th subsystem is the linear system of size $n-1\times n-1$ where the $i$th column of $A$ has been removed.
My question is: suppose the initial solution of $A^TAx=A^Tb$ is componentwise nonnegative (that is, $x\geq 0$), is it always true that at least one of the subsystems has a componentwise nonnegative solution?
Example:
Given the initial normal equations $\begin{pmatrix}9&4&2\\4&5&0\\2&0&1\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}7\\3\\2\end{pmatrix}$, with the componentwise nonnegative solution $x=\begin{pmatrix}\frac{1}{3}\\\frac{1}{3}\\\frac{4}{3}\end{pmatrix}$,
the 3 subsystems are:
- $\begin{pmatrix}9&4\\4&5\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}=\begin{pmatrix}7\\3\end{pmatrix}$, with the solution $\begin{pmatrix}\frac{23}{29}\\\frac{-1}{29}\end{pmatrix}$
- $\begin{pmatrix}9&2\\2&1\end{pmatrix}\begin{pmatrix}x_1\\x_3\end{pmatrix}=\begin{pmatrix}7\\2\end{pmatrix}$, with the solution $\begin{pmatrix}\frac{3}{5}\\\frac{4}{5}\end{pmatrix}$
- $\begin{pmatrix}5&0\\0&1\end{pmatrix}\begin{pmatrix}x_2\\x_3\end{pmatrix}=\begin{pmatrix}3\\2\end{pmatrix}$, with the solution $\begin{pmatrix}\frac{3}{5}\\2\end{pmatrix}$.
We can observe that among the three solutions, there is at least one which is componentwise nonnegative (actually two in this case).
Is this property true for every system of normal equations with componentwise nonnegative solution?
I implemented a code testing this property on random examples, and I was not able to find a counterexample.
Thanks!