Assume i have $n$ 3D unit vectors $v_s$, with different values. Then i define a matrix $T$ as:
$$ T = \frac{1}{n} \sum_{s=1}^{n} v_s \times v_s' $$ where $v_s$ are $3\times 1$ vectors and therefore $T$ is a $3\times 3$ matrix.
How can i find a possible set of vectors $v_s$ if a positive definite diagonal matrix $T$ (with $trace=1$) is given? (any numerical or non-exact method is acceptable)