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In my textbook there is the following nameless theorem:

Let $Q=\sum_{i,j=1}^n a_{ij}X_i X_j$ with $a_{ij}=a_{ji}\in K$ be a quadratic form in $n$ variables over a field $K$ not of characteristic $2$. Then there is a linear transformation of coordinates, such that $Q$ has the form $$ Q = \sum_{i=1}^r a_i X_i^2 \text{ with $a_i \neq 0$ for $i = 1, \ldots, r$ } \, .$$ The invariant $r$ is the rank of the quadratic form $Q$.

Is this Sylvester's law of inertia?

I think it must be...

Qyburn
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1 Answers1

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I'm pretty sure you are talking about Hermite reduction. The construction is not the same as Sylvester; we are not making any distinction between positive and negative diagonal elements, indeed a field cannot be ordered unless it is characteristic zero.

NOTE: yes, in George Leo Watson, Integral Quadratic Forms, pages 17-19, he discusses a specific version of this as being Hermite's method of reduction. Earlier on page 9 he mentions a more general version under the heading "Rational diagonalization" where he does not multiply through in order to always have integer entries.

Best with induction on $n.$ Re-order so that $a_{11} \neq 0.$ Consider $$ a_{11}(x_1 + \frac{a_{12}}{a_{11}} x_2 + \frac{a_{13}}{a_{11}} x_3 + \cdots + \frac{a_{1n}}{a_{11}} x_n )^2 $$

This is $$ a_{11} \left( x_1^2 + 2 \frac{a_{12}}{a_{11}} x_1 x_2 + 2 \frac{a_{13}}{a_{11}} x_1 x_3 + \cdots + 2 \frac{a_{1n}}{a_{11}} x_1 x_n + \mbox{ OTHER} \right) $$ or $$ a_{11} x_1^2 + 2 a_{12} x_1 x_2 + 2 a_{13} x_1 x_3 + \cdots + 2 a_{1n} x_1 x_n + \mbox{ OTHER}_2 $$

Now, subtract this off from the original $Q$ you wrote. What remains is a quadratic form in $n-1$ variables. Repeat the first step.

CAVEAT. I need to think about what happens when the only diagonal entries remaining are zero. We just make a trivial change to force the first diagonal entry to be nonzero. For example, consider the form $2xy$ in just two variables. If we take $x = u-v, \; y = u+v$ the new form is $2 u^2 - 2 v^2.$ That is, the ability to force the first diagonal coefficient to be nonzero is simply the ability for the form to represent a nonzero number, that is the form is not the zero form.

Will Jagy
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  • Concerning what you say in order a field be ordered, you have to add besides the characteristic zero the condition "-1 is not a square". – Piquito Apr 18 '15 at 21:59
  • @LuisGomezSanchez, not actually enough. In $\mathbb Q (\sqrt{-2}),$ $-1$ is not a square, but it is the sum of two squares. http://en.wikipedia.org/wiki/Stufe_%28algebra%29#Examples – Will Jagy Apr 18 '15 at 23:58
  • Hi Will. Are there an order in your field? I'd love to know it. – Piquito Apr 19 '15 at 14:18
  • @LuisGomezSanchez, that example is not an ordered field. $-1$ is not a square but the field is still not ordered. – Will Jagy Apr 19 '15 at 18:37
  • @LuisGomezSanchez "Every ordered field is a formally real field" from http://en.wikipedia.org/wiki/Ordered_field then definition "−1 is not a sum of squares in F." from http://en.wikipedia.org/wiki/Formally_real_field#Alternative_Definitions – Will Jagy Apr 19 '15 at 18:48