Spectral norm of similar matrices

Question

In the table it says that if $$A=P B P^{-1}$$ then spectral norm is the same for

1. Similar matrices.
2. Unitary similar matrices.

Is the first statement (1) true? If $A=P B P^{-1}$ (where $P$ is not a unitary matrix)?

Answer 1

a) is not true: the matrices $$ A=\pmatrix{ 1 & 0 \\ 0 & 2} $$ and $$ B=\pmatrix{ 1 & n \\ 0 & 2} $$ are similar as their Jordan normal form is equal to $A$, but $\|A\|_2 = 2$, $\|B\|_2 \ge n \to \infty$ for $n\to \infty$.

In case you like, here is the explicit transformation: One can check that $A = S^{-1}BS$ with $S=\pmatrix{ 1 & n \\ 0 & 1}$ so $S^{-1} = \pmatrix{1&-n\\0&1}$.