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

Question

It is true that $\rho(A)\le\|A\|_p$ but is there an example where the inequality is strict? ($\rho$ is the spectral radius i.e. $\max|$eigenvalues of $A|$)

Though I understand the definition of a norm of a matrix I don't know how to calculate it.

Answer 1

Take\begin{array}{rccc}A\colon&\mathbb{R}^2&\longrightarrow&\mathbb{R}^2\\&(x,y)&\mapsto&(x+y,y).\end{array}Its only eigenvalue is $1$ and therefore its spectral radius is $1$. But $A(0,1)=(1,1)$ and therefore $\|A\|\geqslant\sqrt2$ (actually, it is an equality).