For my machine learning course, I have to find the SVD of the following matrix $\begin{pmatrix} 2 & 1 \\ 2 & 1 \\ \frac{2}{5} & \frac{11}{5} \\ \frac{2}{5} & \frac{11}{5} \\ \end{pmatrix}$.
I found that $\sigma_{1} = 4$ and $\sigma_{2} = 2$. With those information, I found that the V matrix equals V = $ \begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{-3}{5} \\ \end{pmatrix} $
Finally, I managed to compute $u_{1} = \begin{pmatrix} 0.5 \\ 0.5 \\ 0.5 \\ 0.5 \\ \end{pmatrix} $ and $u_{2} = \begin{pmatrix} 0.5 \\ 0.5 \\ -0.5 \\ -0.5 \\ \end{pmatrix}$.
My question is the following: How can I compute $u_{3}$ and $u_{4}$ in a mathematical way ( I found that $u_{3} = \begin{pmatrix} -0.5 \\ -0.5 \\ 0.5 \\ 0.5 \\ \end{pmatrix} $ and $u_{4} = \begin{pmatrix} 0.5 \\ -0.5 \\ -0.5 \\ 0.5 \\ \end{pmatrix}$ could work)? Should I use the null space on A transpose?
Thanks in advance.
$U_{3}$ and $U_{4}$ aren't uniquely defined- any two orthogonal vectors in the null space of $A^{T}$ will do. You could, for example, compute a basis for $N(A^{T})$, and then apply the Gram-Schmidt process to orthogonalize the basis.
Note that in multiplying out $A=U\Sigma V^{T}$, these last two columns of $U$ are multiplied by the corresponding singular values, which happen to be 0. Thus they don't contribute anything to the product.