My guess is, it only holds one way i.e. $A>B$ then $\sqrt{A}>\sqrt{B}$ but not otherwise. Any proof or counterexample would be appreciated.
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Does A > B means , all the element of Mat A > all the element of Mat B ?? – Shravan40 Oct 16 '13 at 04:55
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$A>B$ refer to partial ordering of SPD matrices. It implies $A-B$ is positive definite – arbitUser1401 Oct 16 '13 at 04:56
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Your guess is true. – user1551 Oct 16 '13 at 06:22