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

by devoured_elysium
Tags: positive, prove, traceaa
 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
 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.)
P: 161
 Quote by devoured_elysium 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.

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

$$\sum_{i,j} A_{ij}(A^*)_{ji}$$

What is the definition of A*?
 P: 161 Ah yes, thank you for the note.

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