
#1
Dec1406, 11:41 PM

P: 15

Hello
I'd like to know how to prove that the trace of A.A* is positive. I don't really know how to handle the imaginary part of it. If A has any complex number in it, is it possible to get traces like 102i? If yes, do I consider it as a positive number or negative? Thanks in advance 



#2
Dec1406, 11:51 PM

Sci Advisor
HW Helper
P: 2,020

First of all, the diagonal entries of AA* are real. You can't really compare two compex numbers like that as there is no order on C.
Now, what does the (i,j)th entry of AA* look like? What about the (i,i)th entry? (Side note: tr(AA*) isn't always positive  it can be zero. So a better thing would be to say that it's nonnegative.) 



#3
Dec1506, 12:46 AM

P: 161





#4
Dec1506, 04:37 AM

Sci Advisor
HW Helper
P: 9,398

How to prove trace(A.A*) is positive
That would not be a very easy way of doing this question. The trace of (AA*) is
[tex] \sum_{i,j} A_{ij}(A^*)_{ji}[/tex] What is the definition of A*? 



#5
Dec1506, 09:21 AM

P: 161

Ah yes, thank you for the note.



Register to reply 
Related Discussions  
Trace  Advanced Physics Homework  5  
Trace  Advanced Physics Homework  8  
How to prove that the positive irrationals are not denumerable?  General Math  19  
More Proofs: Prove that if n is an odd positive int., then n^2 == 1(mod 8)  Calculus & Beyond Homework  6  
need to prove that the following integral is positive  Calculus  5 