Let P and Q be Hermitian positive definite matrices. We prove that for all ,

if and only if .

I suppose we need to show that is positive semidefinite iff is also positive semidefinite.

But I am not sure how to proceed.

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- May 4th 2010, 04:40 PMmath8Positive semidefinite matrices
Let P and Q be Hermitian positive definite matrices. We prove that for all ,

if and only if .

I suppose we need to show that is positive semidefinite iff is also positive semidefinite.

But I am not sure how to proceed. - May 5th 2010, 12:57 AMOpalg
See this thread.

- May 5th 2010, 07:20 AMmath8
Thanks, I read the post and it makes sense but I still have a few questions:

when you say A has a positive square root , do you mean is positive definite?

And why do we need it to be positive? - May 5th 2010, 07:59 AMmath8
Also, how do you prove that if then ?

- May 5th 2010, 01:43 PMOpalg
Yes, that is what it means; and no, for this proof it's not important to take the positive square root of A. In general, a positive definite matrix has many square roots (but only one of them is positive definite), and for this proof any square root would do.

The only way I know to prove that is to use the*norm*of an operator. Define , where . Then and . It follows that

.