Intuition and name for $a^T R a$ I have seen this operation before but cant seem to recall the name or corresponding theory. This operation changes inner product $<a,b> = a^T b$ to $a^T R b$, where $R$ is some matrix. What does this do intuitively and what is the name? 

If $R$ is symmetric and positive definite, then the operation you are referring to is simply a change of basis on the underlying space. Such a change results in the entries of the column matrices representing the vectors $a$ and $b$ changing. We want to keep the inner product unchanged (as the inner product shouldn't depend on what coordinates we impose on the space), so we correct it by inserting $R$. We actually have $R=A^TA$ where $A$ is the change of basis matrix in the appropriate direction.
