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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.

Shiyu

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    @Calle: Yes, you are right. $Id-r_i r_i^T$ in fact is an orthogonal projection matrix with rank 2. Its eigenvalues are 1, 1 and 0. But the sum of some orthogonal projection matrices is a positive definite matrix.2011-02-16

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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 $x^T x\sum (r^T_i x )r_i-\sum(r^T_ix)^2x=0$.2011-02-16