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I was wondering there is an elegant solution for

$$\sum_i J_i \left(J_i^T x - 1 \right) = 0$$

where $x, J_i \in \mathbb{R}^n$ are vectors. Also all $J_i$ have the same norm.

I thought of having such approch

$$\sum_i J_i \left(||J_i|| \cdot ||x|| \cdot cos(\phi_i) - 1 \right) = 0 \\ ||J_i|| \cdot \sum_i J_i \left(||x|| \cdot cos(\phi_i) - 1 \right) = 0 $$

Has anyone thoughts on this? Thanks

I am aware that I can use an Cholesky-factorization if $\sum_i J_i \cdot J_i^T$ is positive-definite but I wanted to try something else.

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    What is the range of variation for index $i$: from $1$ to $k$ with $kn$ ? If $k=n$, you need supplementary assumptions : are the $J_i$ independent or not, etc. – 2017-01-13

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