Here I have an interesting problem on linear algebra. It looks very simple, but not so easy to solve for me.
Let $r_i, i=1,…,n$ be unit vectors in $\mathbb{R}^n$, find a unit vector $x$ to minimize $\sum \| r_i\times x \|^2$
Remark: if let $\theta$ be the angle between $r_i$ and $x$, then $\sum \| r_i\times x \|^2 = \sum \sin^2 \theta _i$. But I don't like sinusoid functions, I think they make the problem more complex especially for high dimensional cases. Is it possible to solve the problem using linear algebra or matrix analysis?
Thank you very much.
Shiyu