Quadratic Forms: Closed Form from Values on Basis?

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SUMMARY

The discussion centers on deriving a closed form for a quadratic form \( q \) defined on \( \mathbb{Z}^4 \) using known values on the basis vectors and correction terms for non-bilinearity. The matrix representation of the quadratic form is expressed as \( q(v) = v^\tau Qv \), where \( Q \) is the matrix to be determined. The key insight is that by treating the entries of \( Q \) as variables, one can formulate equations based on the values of \( q \) at the basis vectors to solve for \( Q \). This approach allows for the computation of \( q \) at any 4-tuple in \( \mathbb{Z}^4 \).

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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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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.
 

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