Quadratic form

Diagonalization of a quadratic form

Away from 2 a quadratic form

can be transformed to

under appropriate change of coördinates where .¹ geo

Proof

Without loss of generality it can be assumed that . For if some then we can permute coördinates. If for all , then we may assume by the same token. Let , , and otherwise for . Then where . Hence we can choose whence .

Under this assumption, it follows

for some . Let and otherwise for . Then

One can then repeat the same steps for &c. until one has a quadratic form

where iff the corresponding quadric is singular.

It follows that A quadric is singular iff its matrix is singular away from 2.

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Footnotes

1. 2020. Finite geometries, ¶4.25, p. 91 ↩