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?
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.
