I just read that any positive definite quadratic form in $n$ variables can be brought to the sum of $n$ squares by a suitable invertible linear transformation, and that geometrically this means that there is only one positive definite real quadratic form of every dimension.
Can somebody help me understand this / expand on this? I have a good understanding of what a positive definite quadratic form in $n$ variables is. Let $Q$ be such a quadratic form. Then is this saying that there is a basis of our vector space such that $Q(x_1,..,x_n)=a_1^2+...+a_n^2$? And then perhaps that each summand can be scaled arbitrarily, so in this sense there is only one positive definite real quadratic form in every dimension. Is this right? Is there anything else I'm missing? Any other interesting factoids lingering close by? Thanks!
You are right in the fact that the quadratic form can be put into diagonal form, but there will be several different real quadratic forms on each rank. Take the example when $n=3$, then you can write your form as $$ a_1x^2+a_2y^2+a_3z^2$$ You can rescale by a change of variables but just removing any squares from the $a_i$. But since in $\mathbb{R}$ you always have $\sqrt{a}$ for any $a>0$ your quadratic form will be one of the following $$x^2+y^2+z^2$$ $$x^2+y^2-z^2$$ $$x^2-y^2-z^2$$ $$-x^2-y^2-z^2$$ The first and last are definite forms, and the middle ones are indefinite. The signature is the number $2i-n$ where $i$ is the number of positive terms, this is an invariant of your quadratic form.