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Let $A$ be a $N\times N$ non-singular real symmetric matrix. Consider the system of equations $$ (x_i - a_i)\sum_j A_{ij} x_j = b_i $$ for $i=1,...,N$. The vectors $a$ and $b$ are arbitrary parameters.

What can one say about the $x$ that solve this system of equations? In the solution unique? Is it exists a close form solution?

I am particularly interested in the case $b_i = b$ for all the $i$.

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Every individual equation describes a (hyper-)quadric, and there can be up to $2^N$ intersections (without degeneracies). By painful elimination, you can probably reduce to a univariate equation of degree $2^N$, which won't have a closed-form solution in general (but for $N=1,2$).

For the sake of illustration, consider $A$ to be a unit matrix, all $a_i=0$ and all $b_i=1$. Then every equation describes a pair of hyperplanes $x_i=\pm1$ and the solutions are the $2^N$ tuples $(\pm1,\pm1,\cdots\pm1)$.

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  • $\begingroup$ If $A$ is the unit matrix and $a_i=0$ the solutions are the 2^N tuples $(\pm b_1,\pm b_2, \dots, \pm b_N)$ $\endgroup$ – JacopoCrickets Dec 18 '17 at 18:05
  • $\begingroup$ @JacopoCrickets: there was a typo $b_i=0$. $\endgroup$ – Yves Daoust Dec 18 '17 at 19:56

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