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Max norm

The max norm is the elementwise norm with $p = \infty$: $ \|A\|_{\text{max}} = \max \{|a_{ij}|\}. $ This norm is not sub-multiplicative.

Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. How are $\|A\|_{\text{max}}$ and $\|D\|_{\text{max}}$ related?

Some examples seem to indicate that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$. Is that all that one can say?

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Yes, that's all you can say. The inequality you mention holds because $\|D\|_{\max}$ is the operator norm of $A$, i.e. the biggest eigenvalue in absolute value.

For extreme examples of the inequality, consider the $n\times n$ matrix $ A=\begin{bmatrix}1&\cdots&1\\ \vdots&\ddots&\vdots\\ 1&\cdots&1\end{bmatrix}. $ Then $ D=\begin{bmatrix}n\\ & 0 \\ & & \ddots \\ & & & 0\end{bmatrix}, $ so $\|A\|_{\max}=1$, $\|D\|_{\max}=n$.