In my statistics notes, I'm given some "useful matrix algebra results". Two of those results are:
$$\boldsymbol{A} \;\text{ is a } p \times p \text{ matrix}$$ $$\boldsymbol{B} \;\text{ is a } p \times 1 \text{ column vector}$$
$$\frac{\partial}{\partial \boldsymbol B}(\boldsymbol {B}^T \boldsymbol {AB})=(\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{B} \;\;\;\;\;\text{ or,}$$ $$\frac{\partial}{\partial \boldsymbol B}(\boldsymbol {B}^T \boldsymbol {AB})= 2\boldsymbol{AB} \;\;\;\;\;\text{ if } \boldsymbol A \text{ is symmetric}$$
I can do some "proofs" by example and so on to convince myself it works, but I don't know how to interpret these two results. Is there a visual interpretation or intuitive interpretation of this, rather than just viewing it as some mechanical matrix algebra?
Side note: do these results have a name? I haven't been able to find these results written anywhere except my statistics notes.
If it's 2 by 2, it's called representing quadratic function in a matrix form, where $B$ is the variables and $A$ with its coefficients. Watch this video for more details.