Let $\mathbf{R}\in\mathbb{R}^{n\times n}$ be upper triangular and $\|\cdot\|$ be the induced 2-norm of matrices. Then, does $\|\mathbf{R}\| \le \|\mathbf{R} + \mathbf{R}^T\|$ hold?
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3False for $n = 3$ and $\mathbf R = \begin{pmatrix} 4 & 0 & 0 \ 32 & 5 & 0 \ 24 & -24 & 9 \end{pmatrix}^T$. – darij grinberg Aug 17 '19 at 06:11
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On the other hand, it looks like it's true for $n = 2$. – darij grinberg Aug 17 '19 at 06:15
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@darijgrinberg Thanks. I was trying to disprove it when $n = 2$ but couldn't. – flyingwith Aug 17 '19 at 06:17
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@darijgrinberg Regarding which matrix norm? – Sudix Aug 17 '19 at 06:29
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2@Sudix: The induced $2$-norm, i.e., the largest singular value. – darij grinberg Aug 17 '19 at 06:32