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Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. I've learned that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$, where $\|A\|_{\text{max}}$ means the Max norm.

I want to get a sharper bound $\|A\|_{\text{max}}$ by using the knwoledge about more than one eigenvalue, let say two. Ky-Fan norms seem appropriate, so I'm looking for something like $ \|A\|_{\text{max}}\not <\frac12\|D\|_2, $ where $\|D\|_2=\lambda_0+\lambda_1$ sums up the largest eigenvalues. Numerics showed that it doesn't hold, ven if I use absoulute values $|\lambda_0|+|\lambda_1|$.

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What you want is impossible, unless you put further restrictions on which $A$ are allowable. To see why, start with $A$ diagonal, so that $D=A$. In this example, your original inequality is sharp. Then any linear combination of the diagonal of $D$ with coefficients less than $1$ in absolute value will fail your desired inequality.

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    Nope, the bound is still sharp for positive definite, take A=D=\begin{bmatrix}2&0\\0&1\end{bmatrix}.2012-11-13