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

1. Dec 14, 2006

### devoured_elysium

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?:zzz:

2. Dec 14, 2006

### morphism

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. Dec 15, 2006

### doodle

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.

4. Dec 15, 2006

### matt grime

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*?

5. Dec 15, 2006

### doodle

Ah yes, thank you for the note.