Proving Symmetry and Definiteness of Bilinear Form q in Real Vector Space V

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In summary: The symplectic form is the decomposition of vector space V using this form. It is a very specific form, and is just the matrix of f with all elements swapped around. It is non-degenerate, which means that for every different vector v in V, there exists a w in V such that f(v,w) is different than zero.
  • #1
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let f:VxV->R be an antisymmetric billinear form in real vector space V, there exists an operator that satisfies J:V->V J^2=-I.
i need to prove that the form q:VxV->R, for every a,b in V q(a,b)=f(a,J(b)) is symmetric and definite positive.
i tried to show that it's symmetric with its definition and that J^2=-I, but with no success, any hints here how to approach this question?
 
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  • #2
If a and b are two vectors, you know, from the formula what q(a,b) are. What is q(b,a), also using the formula. For the positive definite part, you will need to look at q(v,v) for v any vector.
 
  • #3
Aren't you omitting something key about f?
 
  • #4
yes, but i don't how to say it english.
i know that it's equivalent to that the matrix of f is invertible, i.e has an inverse.
does this help to resolve the question?
 
  • #5
halls it doesn't help, i tried it this way but got stuck.
 
  • #6
Google for symplectic forms - that's what this is. The point about f is that you can pick an element, v, and using f and its non-degeneracy, find a w so that f(v,w)=-1...
 
  • #7
yes, f is non degenrate.
but i don't see how does it help, i need to show that q(a,b)=q(b,a)
wait, a minute if f is antisymmetric then it means that f(x,y)=-f(y,x) then f(a,a)=0 always, but i don't see how can i find a w such that f(v,w)=-1, i mean if J^2=-I then if we write w=J(v), then J^2(v)=J(w)=-v
then f(-J(w),J(v))=f(v,J(v))=-f(J(v),v)=f(J(v),J(w))
f(J(v-w),J(v+w))=0, but how procceed from here?
 
  • #8
loop quantum gravity said:
i don't see how can i find a w such that f(v,w)=-1

it's a non-degenerate bilinear form, the very definition means that you can find such a w given a non-zero v. That is almost precisely what non-degenerate means.

Ah, wait - I misread your question - I thought you had to prove that J existed. The point is that that J is not any old J, is it? It is a very particular J, and is gotten by taking the decomposition of V using the symplectic form.
 
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  • #9
yes ofcourse that J is the symplectic form matrix, (and to be quite frank with you, i had to prove it before the question iv'e given here, which i ofcourse did).
i know that to be non degenerates means that for every v different than zero, there exists w in V such that f(v,w) is different than zero, anyway i can choose w to be w=(1,0,0,...,0) and v=(0,1,0,..,0) then f(v,w)=-1, but how does this helps me to prove that q is symmetric and definite positive?
 
  • #10
If all else fails, just use the natural symplectic basis.
 

1. What is a bilinear form in a real vector space?

A bilinear form in a real vector space V is a function q: V x V -> R that takes in two vectors from V and outputs a real number. It is linear in both of its arguments, meaning that q(av, w) = aq(v, w) and q(v1 + v2, w) = q(v1, w) + q(v2, w) for all vectors v, w in V and scalar a.

2. How do you prove that a bilinear form is symmetric?

To prove symmetry of a bilinear form q in a real vector space V, we need to show that q(v, w) = q(w, v) for all vectors v, w in V. This can be done by using the properties of linearity and the definition of a bilinear form. We can also use a counterexample to show that the bilinear form is not symmetric.

3. What is the process for proving definiteness of a bilinear form?

The process for proving definiteness of a bilinear form q in a real vector space V involves showing that for all vectors v in V, q(v, v) is either always positive, always negative, or always zero. This can be done by using properties of linearity and using the definition of a bilinear form. We can also use a counterexample to show that the bilinear form is not definite.

4. Can a bilinear form be both symmetric and definite at the same time?

Yes, a bilinear form can be both symmetric and definite at the same time. This means that q(v, w) = q(w, v) for all vectors v, w in V, and q(v, v) is always positive, negative, or zero. An example of a bilinear form that is both symmetric and definite is the dot product in a real vector space.

5. How is proving symmetry and definiteness of a bilinear form useful?

Proving symmetry and definiteness of a bilinear form is useful because it helps us understand the properties of the form and how it behaves in a real vector space. It also allows us to use the form to solve various mathematical problems and applications, such as finding eigenvalues and eigenvectors, solving optimization problems, and studying quadratic forms.

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