Given a general linear system $Ax = y$ and the bilinear form $z(x,y) = y^T Ax$, what are the links between these two mathematical objects?
Thanks.
EDIT: Original question is too general and difficult to answer, so let me to tell you why I posted it.
In proving SVD theorem, one starts observing that for every matrix $A \in \mathbb{R}^{m \times n}$, by a compactness argument, there exist unitary vectors $u$ and $v$ such that:
$$Av=\sigma u$$ with $\sigma = ||A||_{2}$. Now:
1) May we translate this statement in terms of the above bilinear form $u^T Av$?
2) If yes, how to do it?
Thanks again.