Representation of a quadratic form in terms of vector-matrix norms

Question

There is a quadratic form:

$q=x^T \cdot A \cdot x$

here $x$ - arbitrary $n \times 1$ vector and $A$ - arbitrary $n \times n$ matrix

Is there a representation of this quadratic form in terms of the vector-matrix norms of its components, i.e.:

$x^T \cdot A \cdot x = f(||x||,||A||)$

It should be strict equality, not inequality and not bounds.

EDIT:

I found this thread and took the upper bound for the quadratic form from it: Reference for upper bound $x^T A x \leq \lambda_{\max}(A) \|x\|^2$?

That is, in some cases, the bounds can play the role of equality. But this is not always possible.

Answer 1

No: for example, consider $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$. If $x = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $y = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, then $\|x\| = \|y\| = 1$, yet $1 = x^T A x \neq y^T A y = 2$.