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


by devoured_elysium
Tags: positive, prove, traceaa
devoured_elysium
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#1
Dec14-06, 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 10-2i? If yes, do I consider it as a positive number or negative?

Thanks in advance
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morphism
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#2
Dec14-06, 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.)
doodle
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#3
Dec15-06, 12:46 AM
P: 161
Quote Quote by devoured_elysium View Post
I'd like to know how to prove that the trace of A.A* is positive.
My sketchy knowledge about linear algebra tells me that you would have to relate the nature of the singular values of AA* to its trace being positive.

matt grime
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#4
Dec15-06, 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*?
doodle
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#5
Dec15-06, 09:21 AM
P: 161
Ah yes, thank you for the note.


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