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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.


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What is the cross product in $\mathbb{R}^n$? Does $x=0$ solve your problem? – Thomas Rot Feb 16 '11 at 14:42
The $ \times$ means cross product. See – Shiyu Feb 16 '11 at 14:47
I forgot to add a constraint that $ x $ is a unit vector. – Shiyu Feb 16 '11 at 14:48
@Thomas: @Theo: OP has now called for $x$ to be a unit vector, so $x=0$ won't work, probably because of your comments. I think the best way to view the request is $\sum \| r_i\times x \|^2 = \sum \sin^2 \theta _i=\sum 1-\cos^2 \theta_i=\sum 1-(r_i\cdot x)^2$ – Ross Millikan Feb 16 '11 at 14:49
@Shiyu: but the cross product is difficult to define in dimensions other than 3. This is essentially because a 3x3 antisymmetric matrix has 3 independent components, but a 4x4 has 6, not 4. The general case is n(n-1)/2. That is why I went to the dot product, which is well defined. – Ross Millikan Feb 16 '11 at 14:54
up vote 0 down vote accepted

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.

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thanks. In fact, we can directly calculate the derivative of ${\sum (r^T_i x )^2}/{x^T x}$ with respect to the vector $x$. Let the derivative be zero and we have $x^T x\sum (r^T_i x )r_i-\sum(r^T_ix)^2x=0$. – Shiyu Feb 16 '11 at 15:51

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