Is there any connections between $\max_x \frac{xMx^\top}{xx^\top}$ and $\max_x \frac{xM^kx^\top}{xx^\top}$?

Question

Fix a matrix $M$. Consider the following two quantities: $$ \lambda_1 = \max_{x \in \mathbb{R}^d} \frac{xMx^\top}{xx^\top} $$ $$ \lambda_k = \max_{x \in \mathbb{R}^d} \frac{xM^kx^\top}{xx^\top} $$

Any ideas if there are any connections (inequality?) between these two quantities?

Answer 1

If $M$ is real symmetric and its spectral radius happens to be its largest eigenvalue (note: the spectral radius is the largest modulus of the eigenvalues, not necessarily the largest eigenvalue), then $\lambda_k=\lambda_1^k$, but in general, the two quantities are not related to each other.