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I know that for two symmetric positive semi-definite (non-diagonal) matrices $A,B$, the inequality asserts that the following does not hold for all $p > 1$

$$A \succeq B \succeq 0 \Rightarrow A^p \succeq B^p $$

I would like to know if there are results, which say under what conditions of $A,B$ (e.g., looking at a particular space of the the matrices etc.) the inequality would hold, and in particular, for $p = 2$? Thanks.

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  • $\begingroup$ Since they are symmetric, you can diagonalize $A$. And I think if the minimum of nonzero eigenvalues $\lambda_{min}(A)>1$ it holds. $\endgroup$
    – percusse
    Commented Aug 22, 2016 at 4:47

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