How does one find the reduced Singular Value Decomposition of a row or column vector? If we treat a column vector $a$ as an $n \times 1$ matrix, or a row vector $a^T$ as a $1 \times n$ matrix, how would one write out the reduced singular value decomposition of $a$?
 A: $$a=(a/\| a \|) \begin{bmatrix} \| a \| \end{bmatrix} \begin{bmatrix} 1 \end{bmatrix}.$$
This pretty much follows by knowing the shape of a reduced SVD: if $A \in \mathbb{C}^{m \times n}$ then $U \in \mathbb{C}^{m \times r}, \Sigma \in \mathbb{R}^{r \times r}, V^* \in \mathbb{C}^{r \times n}$, where $r=\text{rank}(A)$. Now take $m=k,r=1,n=1$ (where $a$ is $k \times 1$).
A: I know this is not an answer, but it might be instructive to write out the full SVD:
For a row $a^T$, let $U =1$, $\Sigma =\begin{bmatrix} \|a\| & 0 & \cdots & 0 \end{bmatrix}$,
$V = \begin{bmatrix} {1 \over \|a\|} a^T \\ v_2^T \\ \vdots \\ v_n^T \end{bmatrix}$,
where ${1 \over \|a\|} a^T, v_2, \cdots,\ v_n$ form an orthonormal basis.
For a column $a$, replace $U,\Sigma, V$ by $V^T, \Sigma^T, U^T$ respectively.
A: The truncated SVD of $\mathbf{a} \in \mathbb{R}^{n \times 1}$ would be 
$$
\mathbf{a}
=
\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{T},
$$
where $\mathbf{U} = \mathbf{a} / \|\mathbf{a}\|\in \mathbb{R}^{n \times 1}$, $\mathbf{\Sigma} = [\|\mathbf{a}\|] \in \mathbb{R}^{1 \times 1}$ and $\mathbf{V} =[1] \in \mathbb{R}^{1 \times 1}$.
Essentially, the only non-trivial thing in this decomposition is to normalize the length of $\mathbf{a}$ to obtain $\mathbf{U}$, because the decomposition (at least to the best of my knowledge) requires that $\mathbf{U}$ (and $\mathbf{V}$) have orthonormal columns.
The full SVD could be obtained by setting
$\mathbf{U} = \left[ \frac{1}{\|\mathbf{a}\|}\mathbf{a} \;\; \mathbf{U}^{\prime}\right]\in \mathbb{R}^{n \times n}$, with $\mathbf{U}^{\prime}$ being any $n \times (n-1)$ matrix with orthonormal columns that are also orthogonal to $\mathbf{a}$,
 $\mathbf{\Sigma} = [\|\mathbf{a}\| \; 0\; 0 \; \dots \; 0]^{T} \in \mathbb{R}^{n \times 1}$ and $\mathbf{V} =[1] \in \mathbb{R}^{1 \times 1}$.
