Diagonalization of a quadratic form

can be transformed to

under appropriate change of coördinates where .¹ #m/thm/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.

#state/tidy | #lang/en | #SemBr

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

Diagonalization of a quadratic form

Footnotes