my question may seem particularly broad, but I was curious whether there was any particualar field or application where there exists a relationship between $A$ $*$ $v$ and $A^T*$ $v$, with $A$ being a $n$-square matrix and $v \in R^n$.
Thank you in advance
One example is if $A$ is a rotation, then $A^T$ is the inverse rotation. You can the choose whatever relationship you have between the vectors. For example $Av=x$ and $A^Tv=y$, then $x\cdot v=y\cdot v$. Or $A^2y=x$