Diagonalization of a quadratic form

Away from 2 a quadratic form

𝐹 (𝐗) = 𝑛 \sum 𝑖, 𝑗 = 0 𝑡 𝑖 𝑗 𝑋 𝑖 𝑋 𝑗

can be transformed to

𝐹 (𝐗) = 𝑟 \sum 𝑖 = 0 𝑎 𝑖 𝑌 2 𝑖

under appropriate change of coördinates where $𝑟 \leq 𝑛$ .¹ #m/thm/geo

Proof

Without loss of generality it can be assumed that $𝑡 00 \neq 0$ . For if some $𝑡 𝑖 𝑖 \neq 0$ then we can permute coördinates. If $𝑡 𝑖 𝑖 = 0$ for all $𝑖$ , then we may assume $𝑡 01 \neq 0$ by the same token. Let $𝑋 0 = 𝑌 0$ , $𝑋 1 = 𝑐 𝑌 0 + 𝑌 1$ , and otherwise $𝑋 𝑖 = 𝑌 𝑖$ for $𝑖 > 1$ . Then $𝐹 (𝐗) = \sum 𝑛 𝑖, 𝑗 = 1 𝑠 𝑖 𝑗 𝑌 𝑖 𝑌 𝑗$ where $𝑠 00 = 𝑡 01 𝑐 + 𝑡 10 𝑐$ . Hence we can choose $𝑐 = 𝑡 - 1 01$ whence $𝑠 00 = 2 \neq 0$ .

Under this assumption, it follows

𝐹 (𝐗) = 𝑡 00 𝑋 20 + 𝑋 0 𝑛 \sum 𝑗 = 1 𝑡 0 𝑗 𝑋 𝑗 + 𝑋 0 𝑛 \sum 𝑖 = 1 𝑡 𝑖 0 𝑋 𝑖 + 𝑛 \sum 𝑖, 𝑗 = 1 𝑡 𝑖 𝑗 𝑋 𝑖 𝑋 𝑗 = 𝑡 - 1 00 (𝑛 \sum 𝑖 = 0 𝑡 𝑖 0 𝑋 𝑖) (𝑛 \sum 𝑗 = 0 𝑡 0 𝑗 𝑋 𝑗) + 𝑛 \sum 𝑖, 𝑗 = 1 𝑡' 𝑖 𝑗 𝑋 𝑖 𝑋 𝑗

for some $𝑡' 𝑖 𝑗$ . Let $𝑋 0 = 𝑌 0 - 𝑡 - 1 00 \sum 𝑛 𝑗 = 1 𝑡 1 𝑗 𝑋 𝑗$ and otherwise $𝑋 𝑖 = 𝑌 𝑖$ for $𝑖 > 0$ . Then

𝐹 (𝐗) = 𝑡 00 𝑌 20 + 𝑛 \sum 𝑖, 𝑗 = 1 𝑡' 𝑖 𝑗 𝑌 𝑖 𝑌 𝑗 = 𝑡 00 𝑌 20 + 𝐹' (𝐘)

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

𝐹 (𝐘) = 𝑟 \sum 𝑖 = 0 𝑎 𝑖 𝑋 2 𝑖

where $𝑟 < 𝑛$ 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