Quadratic Forms: Closed Form from Values on Basis?

In summary, the conversation is about a quadratic form q defined on Z<sup>4</sup> and the possibility of finding a closed form or matrix for q that would allow for computing its value at any 4-ple of Z<sup>4</sup>. The suggestion is to treat the matrix Q as variables and use the equations from the value of q on the basis vectors.
  • #1
Bacle
662
1
Hi, Everyone:

I have a quadratic form q, defined on Z<sup>4</sup> , and I know the value of

q on each of the four basis vectors ( I know q is not linear, and there is a sort

of "correction" for non-bilinearity between basis elements , whose values --on

all pairs (a,b) of basis elements-- I do know ).

Question: Is there a way of giving a closed form for q (preferably as a matrix)

that would allow me to compute the value at any 4-ple of Z<sup>4</sup>?

Thanks for any Suggestions, etc.
 
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  • #2
The matrix of a quadratic form is given by ##q(v)=v^\tau Qv##. Treat ##Q## as variables and you get all necessary equations from the value of ##q## on the basis vectors.
 

FAQ: Quadratic Forms: Closed Form from Values on Basis?

What is a quadratic form?

A quadratic form is a mathematical expression that involves variables raised to the second power. It can also be thought of as a polynomial of degree two in multiple variables.

What is a closed form for a quadratic form?

A closed form for a quadratic form is an expression that represents the quadratic form in terms of a finite number of operations such as addition, subtraction, multiplication, and division. It does not involve any variables or unknowns.

How can you determine a closed form for a quadratic form?

To determine a closed form for a quadratic form, you need to know the values of the quadratic form on a basis. Then, using these values, you can solve for the coefficients of the closed form expression using linear algebra techniques such as matrix inversion.

What is a basis for a quadratic form?

A basis for a quadratic form is a set of linearly independent vectors that span the vector space in which the quadratic form is defined. These vectors can be used to represent any point in the vector space in terms of their linear combinations.

Why is finding a closed form for a quadratic form useful?

Finding a closed form for a quadratic form can be useful in solving optimization problems, finding extrema, and simplifying calculations involving the quadratic form. It also allows for easier manipulation and analysis of the quadratic form.

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