Searching $A$ to maximize $\|x\|$ whilst fulfilling a constraint $x^TA^TAx=c$

Question

May $A \in \mathbb R^{n\times n}$ be a symmetric matrix, $x\in \mathbb R^n$ be a vector and $c\in \mathbb R$ be a constant, such that: $$ \langle Ax, Ax \rangle = c $$

If you want to maximize $\|x\|$, how would you choose $A$?

Answer 1

Since your $A$ is real and symmetric, it is diagonalizable and only has real eigenvalues.

Thus we can assume w.l.o.g. that $A$ is a diagonal matrix.

($x^TA^TAx = x^TAAx= x^TS^{-1}DDSx = y^T DD y = y^T D^T D y$)

Hence we can just look at the sum: $$ \langle Ax, Ax\rangle = \sum_{i=1}^n (\lambda_i x_i)^2 $$ Where $\lambda_i, i=1,\dots,n$ are the eigenvalues of $A$.

So you can modify the eigenvalues of $A$ to keep the constraint.

Since if $\max_{i\in[n]}|\lambda_i|$ goes to zero, $||x||$ has to go to infinity.