# [SOLVED]A function being Positive

#### Sudharaka

##### Well-known member
MHB Math Helper
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

##### Well-known member
MHB Math Helper
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.