Bounding lower triangular perturbation

Question

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral norm (or the largest singular value) of a matrix $X\in R^{n\times n}$ 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 $\epsilon$?

Answer 1

I asked a general version of this question here:log norm inequality for lower triangular part of matrix

Unfortunately, I have not received a hint on how to prove it yet, but assuming the problem given in the book there is correct, I use that result. Since $1-\epsilon \leq ||A||_2 \leq 1+\epsilon \Rightarrow ||L||_2 \leq \log_2(2 n (1+\epsilon))$. The result seems correct based on testing a few random matrices in matlab.