Can we express the scalar product $\langle u,Av\rangle$ in terms of $\langle u,v\rangle$?

Question

Given two vectors $u$ and $v$ and a matrix $A$, does anyone see if we can write the scalar product $\langle u,Av\rangle:=u^TAv$ in terms of $\langle u,v\rangle:=u^Tv$? Thanks a lot!

Answer 1

I mean if $A$ is a symmetric positive definite matrix then $<u,Av> = <u,v>_A$ forms a valid norm. But I do not fully understand the question you are asking.

Surely if you take $\tilde{v} = Av$ then $<u,Av> = <u,\tilde{v}>$ which kind of does the job?