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

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?

Thanks in advance

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.)

doodle

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.

matt grime

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

Ah yes, thank you for the note.

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

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

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

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

morphism Recognitions: Homework Help Science Advisor	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.)

matt grime Recognitions: Homework Help Science Advisor	*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*?