Classifying Symmetric Quadratic Forms

In summary, classifying symmetric quadratic forms involves determining whether a given form is positive definite, negative definite, or indefinite. This is done by examining the eigenvalues of the associated matrix and using the Sylvester's Law of Inertia. A positive definite form has all positive eigenvalues, a negative definite form has all negative eigenvalues, and an indefinite form has both positive and negative eigenvalues. Additionally, a quadratic form can also be classified as semidefinite if it has at least one zero eigenvalue. This classification is important in various applications such as optimization problems and differential equations.
  • #1
Bacle
662
1
Hi, All:

I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space
over the reals.

My idea is to use the standard basis for R^3 , then use the matrix representation M
=x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M.

So it seems all symmetric bilinear forms are just all diagonal matrices; maybe we
need to factor out those that are equivalent as bilinear forms, i.e., B,B' are equivalent
if there is a linear isomorphism L:R^3-->R^3 with B(v)=B'(L(v)).

Is this right? Is there anything else?
 
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  • #2
You can even do better! You can even diagonalize those forms such that the diagonal matrix only has 1, -1 or 0 as diagonal entries!
A symmetric bilinear form with 0 as diagonal entry are exactly the non-degenerate forms.

All this actually follows from Gram-Schmidt orthogonalization.

See http://mathworld.wolfram.com/SymmetricBilinearForm.html
 

Related to Classifying Symmetric Quadratic Forms

1. What is a symmetric quadratic form?

A symmetric quadratic form is a mathematical expression that contains only quadratic terms (terms with degree 2) and is symmetric, meaning that the variables can be rearranged without changing the overall expression.

2. How are symmetric quadratic forms classified?

Symmetric quadratic forms are classified based on their discriminant, which is the expression under the square root in the quadratic formula. The discriminant can be positive, negative, or zero, and this determines the type of quadratic form.

3. What are the three types of symmetric quadratic forms?

The three types of symmetric quadratic forms are positive definite, negative definite, and indefinite. Positive definite forms have a positive discriminant, negative definite forms have a negative discriminant, and indefinite forms have a discriminant of zero.

4. How can symmetric quadratic forms be used in real-life applications?

Symmetric quadratic forms can be used to model real-life phenomena such as motion, optimization problems, and economic systems. They are also used in fields such as physics, engineering, and economics to analyze data and make predictions.

5. What is the importance of classifying symmetric quadratic forms?

Classifying symmetric quadratic forms allows for a better understanding of their properties and behavior. It also helps in solving equations involving these forms and in identifying which type of form is most suitable for a given problem or application. Additionally, the classification of symmetric quadratic forms plays a crucial role in fields such as number theory and algebraic geometry.

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