Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator.
Does the following hold?
If $\rho (A) < \rho(B)$ then:
$ || \mathbf{A} \mathbf{w} || < || \mathbf{B} \mathbf{w} || $.
If yes, why?