I always get confused by different notations in linear algebra. For example, I find that this is clear and I can follow everything that is going on by the rules of matrix multiplication
$$ A = U \Sigma V^T $$
But sometimes I see a notation like this...
$$ A = U \Sigma V^T = u_1 \sigma_1 v^T_1 + u_1 \sigma_1v^T_1 + ... + u_r \sigma_r v^T_r $$
and I find it really confusing to understand by looking because its not immediately obvious to me if the subscripts are columns or rows and if the result of these additions are vectors or matrices... How can I read the notation on the right of this equation?
Even worse is the summation notation which makes it even harder to see and would look something like this...
$$ A_{ij} = \sum \sum \sum \text{confusing}_{ij} \text{subscript}_{ji} \text{hell}_{ijk} $$
For the SVD, you can read it as
$$ A = U \Sigma V^{T} = \sum_{i=1}^{r} \sigma_{i} u_{i} v_{i}^{T} $$
as a matrix $A$ is actually the sum of $r$ rank $1$ matrices $u_{i}v_{i}^{T}$ which have a scaling factor of $\sigma_{i}$.
The matrix $u_{i}v_{i}^{T}$ is the matrix generated as the outer product of the two vectors $u_{i} , v_{i}$