Strict inequality in spectral radius estimate

Question

I am interested to find an example which shows strict inequality in the spectral radius estimate that is $r(x) \leq \|x \|$ . I would like to see when $r(x) < \|x\|$.Normal operator is the case where equality holds. Trying to find where the strict inequality hold. Thank you for your help.

Answer 1

Prahlad Vaidyanathan gave a simple example $$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$ to which I'll add an example with $0<r(x)<\|x\|$: $$\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1/2\end{pmatrix}$$