Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. \begin{equation} A=I+L \end{equation} All diagonal entries of $L$ are equal to $0$, so that, all diagonal entries of $A$ are equal to $1$.
Define spectral norm (or the largest singular value) of a matrix $X\in R^{n×n}$ as \begin{equation} \|X\|=sup_{\|v\|_{\ell_2}=1}\|Xv\|_{\ell_2}. \end{equation} Suppose $|\|A\|−1|<\epsilon$. What can I say about $\|L\|$? Is there an upper bound on $\|L\|$ in terms of $\epsilon$?