Tassus Posted September 30, 2011 Posted September 30, 2011 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!
Schrödinger's hat Posted September 30, 2011 Posted September 30, 2011 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.
ajb Posted September 30, 2011 Posted September 30, 2011 I guess you could try doing an explicit example with 2x2 matrices. That would help you.
Tassus Posted September 30, 2011 Author Posted September 30, 2011 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.
Schrödinger's hat Posted October 1, 2011 Posted October 1, 2011 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? 1
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