prove a bound for 2-norm of Jordan block

Question

Let J be a Jordan block.$$J=\begin{bmatrix}x&\epsilon&\\&x&\epsilon&\\&&\ddots&\ddots\\&&&x\end{bmatrix}$$ I recently find out that many J satisfy $||J||_2\leq|x|+\epsilon$, So I wonder if this inequality could be proved stictly or if there exists a counterexample.

Answer 1

As it stands, the statement is false. Consider $x=-\epsilon=1$ for instance.

Your $J$ is equal to $xI+\epsilon N$, where $N$ is the nilpotent Jordan block. Since $N$ has mutually orthogonal columns and all nonzero columns of $N$ are unit vectors, $\|N\|_2=1$. Therefore $$ \|J\|_2=\|xI+\epsilon N\|_2\le |x|\|I\|_2+|\epsilon|\|N\|_2=|x|+|\epsilon|. $$