Let $D\in \mathbb{R}^{n\times n}$ denote a lower triangular matrix. With $\|\cdot\|$ denoting the spectral matrix norm, is there an estimate like


$$
\|D\| \leq C\|D+D^T\|,
$$


where $C>0$ is independent of the dimension $n\in\mathbb{N}$ and $D$?



Norm of the upper triangular part of symmetric matrix
No. The best constant is $C\_n\sim \ln n$. See, for example, [this paper.](https://www.jstor.org/stable/2589115) In particular, for the lower bound, see example 3.3 of the paper.


