Example of a matrix satisfying $\rho(A) < \|A\|$

Question

Give an example of a matrix $A \in \mathbf{C}^{n \times n}$ such that for all operator norms $\| \cdot \|$, one has $\rho(A) < \|A\|$.

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$$\rho(A) := \max \left\{\left|\lambda_{1}\right|, \ldots,\left|\lambda_{n}\right|\right\}$$

where $\lambda_1, \dots , \lambda_n$ are the eigenvalues of $A$. I thought about it for a long time, but unfortunately I could not find an example.

Answer 1

You can take, say, $A=\left[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right]$. Then $\rho(A)=1$, but $A.(0,1)=(1,1)$ and therefore $\|A\|\geqslant\sqrt2>\rho(A)$.

Answer 2

As usual, the canonical example is $$A=\begin{bmatrix}0&1\\0&0\end{bmatrix}.$$ You have $\rho(A)=0$, so $\rho(A)<\|A\|$ for any norm.