The title says it all... I can't see the difference between $a \cdot a^T$ and $a^2$, when $a$ is a vector. However I encountered a formula stating $$\frac{1}{|y+a|} = \frac{1}{|y|} - \frac{y \cdot a }{|y|^3} + \frac{1}{2} \frac{y^T \cdot (3\cdot a \cdot a^T - |a|^2) \cdot y}{|y|^5} + \cdots$$ where $a$ and $y$ are again vectors (Taylor-expansion).
Why is the term in the brackets not simply $ 2 \cdot a^2$?
Ok, I got it now. The term in the brackets is to be interpreted as a matrix $M$ like
$$M_{ij} = 3 a_i a_j - \delta_{ij} |a|^2$$
This makes a lot more sense. But I am not quite sure how to distinguish that from the ordinary dot product...