Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose we have two matrices $A$ and $B$ in which $A \preceq 0$ and $B \succeq 0$. If $\operatorname{tr}(AB) = 0$, can we conclude $AB = 0$ ? Say, $A$ and $B$ are all symmetric.

Thanks a lot!

share|improve this question
    
There is really no need to say A B are all symmetric –  yaoxiao Feb 16 '12 at 23:50
add comment

1 Answer 1

up vote 3 down vote accepted

Yes. WLOG we can assume $A$ is diagonal. Then $\text{tr}(AB) = \sum_j a_{jj} b_{jj}$. Since $a_{jj} \le 0$ and $b_{jj} \ge 0$, the only way this can be $0$ is all $a_{jj} b_{jj} = 0$. But if $B$ is positive semidefinite and $b_{jj} = 0$, we must have $b_{jk} = 0$ for all $k$, so $(AB)_{jk} = a_{jj} b_{jk} = 0$.

share|improve this answer
    
Got you. Thanks a lot! –  mining Feb 17 '12 at 0:07
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.