<s>[INST] Notation - matrix calculatoin i am self-studying linear algebra, and came across the following statement about singular value decomposition (X = $USV^T$). I am somewhat confused on how to interpret the (|) and (--) symbols. Should i interpret that X is the sum of r matrices, of which the first one is equal to the (scalar?) $\sigma_1$ multiplied by the row vector $U_1$ multiplied by the column vector $V_1^T$?    

 [/INST] In the equation, $u_1$ is an $n\times 1$ matrix (i.e., a column vector) and $v_1^t$ is a $1 \times n$ matrix, i.e., a row vector. The product shown is an "outer product", i.e., the result is an $n \times n$ matrix. The little bars are indeed meant to indicate the "shape" of the iterm, i.e., that $u_1$ consists of a bunch of things organized vertically.
To give a concrete example, if 
$$
X = \begin{bmatrix} -5 & -7 \\ 5 & -1\end{bmatrix},
$$ then the SVD is 
\begin{align}
X 
&= \begin{bmatrix} -5 & -7 \\ 5 & -1\end{bmatrix}\\
&= \begin{bmatrix} -3\frac{\sqrt{10}}{10} & \frac{\sqrt{10}}{10} \\ \frac{\sqrt{10}}{10} & 3\frac{\sqrt{10}}{10}\end{bmatrix}
 \begin{bmatrix} 4\sqrt{5} & 0 \\ 0 & 2\sqrt{5} \end{bmatrix}
\begin{bmatrix} \frac{\sqrt{2}}{2} &  \frac{\sqrt{2}}{2} \\  \frac{\sqrt{2}}{2} &  -\frac{\sqrt{2}}{2}\end{bmatrix}\\
&= \begin{bmatrix} -3& 1 \\ 1 & 3\end{bmatrix}
 \begin{bmatrix} 2& 0 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} 1 &  1 \\  1 &  -1\end{bmatrix}
\end{align}
The claim is then that $X$ can be rewritten as
\begin{align}
X &= 
2\begin{bmatrix} -3 \\ 1 \end{bmatrix}
\begin{bmatrix} 1 & 1 \end{bmatrix} +
1\begin{bmatrix} 1 \\ 3 \end{bmatrix}
\begin{bmatrix} 1 & -1 \end{bmatrix}
\end{align}
which indeed turns out to be true. In this case, $u_1 = \begin{bmatrix} -3 \\ 1 \end{bmatrix}$ and $v_1^t = \begin{bmatrix} 1 & 1 \end{bmatrix}$, and so on. 
</s>