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Let $X$ be a random positive semidefinite matrix, and let $A$ be a fixed positive definite matrix. Then, $\forall A$, $$ Pr[X\geq A]\leq Tr(E(X)A^{-1}) $$

Here $X\geq A$ means $A -X$ is positive semidefinite.

Jean Marie
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seeyoutorrm
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  • You can use the term "Loewner order" for this order relationship. – Jean Marie Oct 08 '17 at 06:58
  • Besides, this is the raw question. What have you attempted ? For example, is it true in dimension 1 ? – Jean Marie Oct 08 '17 at 06:58
  • Yes, it is true for sure. I just didn't know how to prove it. – seeyoutorrm Oct 08 '17 at 07:19
  • You mean that you don't know how to treat the elementary case of dimension 1 where it is no longer matrices, but numerical quantities ? – Jean Marie Oct 08 '17 at 07:22
  • I know how to treat the elementary case. But, when it comes to matrix, I cannot handle. – seeyoutorrm Oct 08 '17 at 07:40
  • I think that a natural extension of the dimension 1 case is by working with diagonal matrices, otherwise said, work with diagonalization relationships; I am not certain that it's the best way but this what I would attempt... – Jean Marie Oct 08 '17 at 07:49
  • could you please give some more details or whole proof – seeyoutorrm Oct 08 '17 at 17:18
  • I am sorry. What I said was just a possible direction. I don't know enough random matrices theory. – Jean Marie Oct 08 '17 at 17:35
  • Something puzzles me : the $X$ on the LHS is variable, OK. But the $X$ on the RHS, because one takes a mathematical expectation, has a completely neutral role, and surely has nothing to do with the $X$ in the LHS !!! Could you check the text of your homework ? – Jean Marie Oct 08 '17 at 17:40
  • You haven't reacted to my question: what is the fixed value of $E(X)$ ? – Jean Marie Oct 10 '17 at 22:23

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