Is the Function f Non-Singular and Positive?

[Sudharaka]

Hi everyone, :)

I just want to know what "positive" means in this context? Is it positive definiteness or something? I mean, we cannot speak about the positivity or the negativity of the function \(f\) since it gives out complex values.

Problem:

Prove that the function \(f:\, M_{n}(\mathbb{C})\times M_{n}(\mathbb{C})\rightarrow \mathbb{C}\) given by \(f(X,\,Y)=\mbox{Tr }(X^{t}\overline{Y})\) is non-singular. Is \(f\) positive?

 

[Sudharaka]

Hi everyone, :)

I just want to know what "positive" means in this context? Is it positive definiteness or something? I mean, we cannot speak about the positivity or the negativity of the function \(f\) since it gives out complex values.

Problem:

Prove that the function \(f:\, M_{n}(\mathbb{C})\times M_{n}(\mathbb{C})\rightarrow \mathbb{C}\) given by \(f(X,\,Y)=\mbox{Tr }(X^{t}\overline{Y})\) is non-singular. Is \(f\) positive?

Hi again, :)

I found the answer. This kind of sesquilinear (or Hermitian) bilinear function is called positive when \(f(X,\,X)>0\) for any \(X (\neq 0)\in M_{n}\) which makes sense. :)