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Given a p.s.d matrix $K$, is $2\operatorname{Diag}(K)-K$ a p.s.d matrix? Here, $\operatorname{Diag}(K)$ is a diagonal matrix whose diagonal is the diagonal of $K$.

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No, not necessarily. Define an $n\times n$ matrix $K$ by $K_{ij} = 1$ for every $1\leq i,j\leq n$. $K$ is a positive semi-definite matrix. However, $2I - K$ is not positive semi-definite.

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Is $2Diag(K)-K$ diagonally dominant? –  qlinck Nov 14 '12 at 16:55
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No! All diagonal entries are equal to 1; non-diagonal entries add up to $-(n-1)$ in every row and column. If it was diagonally dominant, it would be psd. –  Yury Nov 14 '12 at 19:10
    
@Yury: Your $2I-K$ is positive semi-definite when $n=2$ :-D –  user1551 Nov 15 '12 at 12:52

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