I see a lot of textbooks trying to maximise $(A^T)BA$, where $A$ is a vector and $B$ is an $n \times n$ matrix.
I tried looking up in a few textbooks, but all I find is an assertion that minimising distance implies maximising the above
What does it do? what does it mean?
Thank you
You may want to take a look at linear and quadratic approximation and quadratic programming. $$ \mathbf{a}^T B \mathbf{a} $$ is called a quadratic form. Since you mention minimization, what you may have seen is the objective function for quadratic programming $$ \mathbf{a}^T B \mathbf{a} + \mathbf{c}^T \mathbf{a} $$ which is minimized subject to some possible constraints such as $M\mathbf{a} \leq \mathbf{b}$. For example, the first couple of terms of a Taylor expansion (2D) produce the Hessian quadratic form: $$ f(\vec{x} + \Delta \vec{x}) \simeq f(\vec{x}) + \nabla^T f(x) \Delta \vec{x} + \frac{1}{2} \Delta \vec{x}^T H \Delta \vec{x} + \dots $$