MHB Is the Function f Non-Singular and Positive?

  • Thread starter Thread starter Sudharaka
  • Start date Start date
  • Tags Tags
    Function Positive
Sudharaka
Gold Member
MHB
Messages
1,558
Reaction score
1
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?
 
Physics news on Phys.org
Sudharaka said:
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. :)
 
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...
Back
Top