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
You want to maximize $\sum_i (r_i \cdot x)^2$ over unit vectors $x$. To get rid of the constraint, this is $$\frac{\sum_i (r_i \cdot x)^2}{|x|^2}=\frac{\sum_i (r_i \cdot x)^2}{\sum_j x_j^2}=\frac{\sum_{ij} (r_{ij} x_j)^2}{\sum_j x_j^2}$$ where $r_{ij}$ is the $j^{th}$ component of $r_i$. Now you can differentiate with respect to $x_j$ and set to zero without any trig functions getting in the way.