The limit of a matrix norm divided by a power of its spectral radius

Question

Let $M$ be a quadratic matrix with spectral radius $\lambda$.

It is known that $\lambda=\lim_{n \to \infty} (||M^n||)^{1/n}$.

I am now interested in the limit of $\frac{||M^n||}{\lambda^n}$ as $n$ goes to infinity. Is there any theorem that leads me from the known fact to the solution of this?

Thanks in advance!

Answer 1

There can be different cases. For example, if $M$ is the identity matrix, then the limit is $1$. If $M=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, then $\lambda=1$, $M^n=\begin{pmatrix}1&n\\0&1\end{pmatrix}$, and so the limit is $\infty$. In general, the limit (if it exists at all) can be any number form $1$ to $+\infty$.