a) Prove using Schur decomposition $\lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1$ where A is in $\mathbb{C}^{mxm}$ and $p$ is the spectral radius.
b) $\lim_{t \rightarrow \infty}||e^{At}||=0 \iff a(A)<1$ where A is in $\mathbb{C}^{mxm}$ and $a$ is the spectral abscissa.
My thoughts are as follows. For (a), I think that $\lim_{n \rightarrow \infty}||A^n||=0$ will only happen if the matrix A is nilpotent. But if the matrix is nilpotent then wouldn't $p(A)=0$ which is less than 1 but I feel like I am missing something.
