Theorems from Spectral applications

For every eigenvalue,

If

then, for every non-negative integer $k$,

Suppose $A=PDP^{-1}$ is diagonalisable, so it has an eigenbasis. A mode contribution $c\lambda^kv$ with $c\ne0$ decays to zero when $|\lambda|<1$, has constant magnitude when $|\lambda|=1$ and grows when $|\lambda|>1$. Without an eigenbasis, a Jordan block can produce additional factors such as $k\lambda^{k-1}$, so the diagonalisation hypothesis is what makes this scalar classification complete (especially at the unit circle).