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A very basic question but i couldn't find another post about it:

Given $p$ non parallel hyper-plane in $\mathbb{R}^p$:

$\left(\begin{array}{cccc} c_{11} & a_{11} & .... & a_{1p} \\ ... & ... & ... & .... \\ c_{p1} & a_{p1} & .... & a_{pp} \end{array} \right)$

$||a_{i.}||=1$

What is the best (from a numerical standpoint of view) way to get the $p$ vector of coordinates $x$ of their intersection?

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If the matrix A is non-singular, the intersection of the hyperplanes is simply the the solution of the linear system of equations $Ax=b$, where

$A= \left(\begin{array}{cccc} a_{11} & .... & a_{1p} \\ ... & ... & .... \\ a_{p1} & .... & a_{pp} \end{array} \right)$

and

$b= -\left(\begin{array}{cccc} c_{11} \\ ... \\ c_{p1} \end{array} \right)$

The "best" way to solve this system largely depends on the matrix A itself (see this , this, or this for more information on a choice of solver).

In the case when A is singular, we cannot describe the intersection as a single point.

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