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How to prove trace(A.A*) is positive 
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#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

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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

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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.



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