Reduce quadratic form of 3 variables to sum

nastupnat0hh

nastupnat0hh

Answered question

2022-03-30

Reduce quadratic form of 3 variables to sum of 3 squares
I am dealing with following quadratic form: q:R3R
q(x,y,z)=xy+yz+xz

Answer & Explanation

Drahthaare89c

Drahthaare89c

Beginner2022-03-31Added 19 answers

As Ben Grossmann said, q is not a sum of squares, but it can always be written as iαiφi2 where αi0 and φi is a linear form.
Nevertheless, you are in the nasty case (the case where there is no squared terms). The trick is to choose two variables (say x and y), isolate all the terms containing at least one of the two variables , write it under the form A[xy+B(z)x+C(z)y], where A is a nonzero constant, and B,C are terms in z only, then "complete the product", and use the identity uv=d14((u+v)2(uv)2).
For you example q(x,y,z)=1.xy+z.x+z.y=(x+z)(y+z)z2=d14(x+y+2z)2d14(xy)2z2
This approach may be generalized to an arbitrary number of variables.

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