Given that $A$ is symmetric $nxn$ matrix.
Show that $\lim_{k \rightarrow \infty} (x^tA^{2k}x)^{1/k}$ exists for all $x \in R^n$ and possible limit values are the eigenvalues of A.
Since A is symmetric you can find an orthonormal basis. So A is similar to the diagonal matrix B. So you get $\lim_{k \rightarrow \infty} (\sum_{i=0}^n x_i^2 \lambda_i^{2k})^{1/k}$. So I can show that this last thing is bounded above and below, but it isn't monotone in k, so how should I show that it's convergent and find what it could possibly converge to?