Hi, I would like some help with this,

 

Assume A and B nxn matrices

 

1. If A,B are positive definite matrices what about the sign of tr(AB)?

 

2. If A is positive definite and B is positive semi definite what about the sign of tr(AB)?

 

Thanks!

Hi, I would like some help with this,

 

Assume A and B nxn matrices

 

1. If A,B are positive definite matrices what about the sign of tr(AB)?

 

2. If A is positive definite and B is positive semi definite what about the sign of tr(AB)?

 

Thanks!

 

This looks a lot like a homework question. We generally don't provide complete answers to such things.

Perhaps you could explain what you've done so far? People will be more inclined to help then.

I guess you could try doing an explicit example with 2x2 matrices. That would help you.

This looks a lot like a homework question. We generally don't provide complete answers to such things.

Perhaps you could explain what you've done so far? People will be more inclined to help then.

 

 

Basically what I want to do, is show that anexpression that involves traces is bounded. A is a Positive definite matrix and B is either positive definite or positivesemi definite (it depends on the assumptions of the other matrices that included in B, which I it set so for convenience). Ithink I can show it by the fact that tr(AB)>0 or tr(AB)>=0.

 

Basically what I want to do, is show that anexpression that involves traces is bounded. A is a Positive definite matrix and B is either positive definite or positivesemi definite (it depends on the assumptions of the other matrices that included in B, which I it set so for convenience). Ithink I can show it by the fact that tr(AB)>0 or tr(AB)>=0.

 

Another way you might be able to approach it:

 

What do you know about about the eigenvalues of a positive-definite or semi-definite matrix?

How do the eigenvalues relate to the trace?

 

Perhaps writing an arbitrary vector as a sum of eigenvectors might help?