Given positive definite $X\in\mathbb{R}^{4\times 4}$, I want to find $Y\in\mathbb{R}^{4\times 2}$, such that $YY'\approx X$.
My attempt:
Using SVD, $X=U\Sigma U^*$. Let $U_i$ be $i'$th column of $U$, then $Y=[U_1 \quad U_2]\begin{bmatrix} \sigma_1 & \\ & \sigma_2 \end{bmatrix}.$
I am not sure if this is a correct approach. Assumption: $\sigma_1\geq \sigma_2\geq \sigma_3\geq \sigma_4$. I think if $\sigma_3$ and $\sigma_4$ are very small compared to $\sigma_1$ and $\sigma_2$ then it might work???