I have an expression like $a^T M b$ which I want to lower bound to something like $a^T M a + ?$ ($a^T M b \geq a^T M a + ?$) where $M \in \mathbb{R}^{nxn}$ is symmetric and positive definite. I can add more assumptions to make this work.
My first idea was to make the sensible assumption $\lVert a-e \rVert \leq \lVert b \rVert \leq \lVert a+e \rVert$ and use this to get a bound which depends in some way on $\lVert e \rVert$ but I am not sure where to start here.
How do I use the lower bound on the norm of the vector to bound my expression?
PS: $a$ and $b$ are actually functions of x but that shouldn't matter here.