I'm a student, currently studying about quadratic forms over a field $\mathbb{R}$ and I have a few questions regarding the topic.

  1. From a book I currently read, a quadratic form is a real-valued function over a vector space $E$ (i.e. $Q: E \longrightarrow \mathbb{R}$, please correct me if I'm mistaken) such that there exists a symmetric bilinear form $B: E \times E \longrightarrow \mathbb{R}$ in which the following expression is valid: \begin{align} Q(x)=B(x,x) \end{align} $\forall x \in E$.

My question: given some function $F: E \longrightarrow \mathbb{R}$. The definition requires the existence of a symmetric bilinear form $G: E \times E \longrightarrow \mathbb{R}$ in which the above expression valid, but how to make sure that we could have such function? For example, if I have a function $F: \mathbb{R^3} \longrightarrow \mathbb{R}$ such that \begin{align} F(x)= x_1+x_2+x_3 \end{align} with $ x=(x_1,x_2,x_3) \in \mathbb{R^3}$,

how to check whether $F$ is a quadratic form or not?

  1. The book also mentioned a regular quadratic space $(E, Q)$, i.e. a vector space $E$ in which it has a quadratic space $Q$ that is nonsingular. What is the definition of nonsingular in terms of function/transformation? And how to connect it to this?

I'm really lost, I know I'm still learning, and I need lots of help. Of course, this is not the last time I'll ask, maybe I'll come again if I find more difficulties but for now, this is all I've got to ask you good people in this community. Thanks! Any help will do!

  • $\begingroup$ The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms. $\endgroup$ – amd Dec 2 '18 at 3:18
  • $\begingroup$ Show that for $E$ finite dimensional with basis $(e_i)$ then $q(\sum_i x_i e_i) = \sum_i \sum_j \frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^\top \frac{A+A^\top}{2} y, q(x) = x^\top \frac{A+A^\top}{2} x$ $\endgroup$ – reuns Dec 2 '18 at 4:01
  • $\begingroup$ @amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then? $\endgroup$ – 20gobbledigook08 Dec 2 '18 at 13:41
  • $\begingroup$ @reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though! $\endgroup$ – 20gobbledigook08 Dec 2 '18 at 13:50
  • $\begingroup$ en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms $\endgroup$ – reuns Dec 2 '18 at 22:16

A function $F:E\to\Bbb R $ is a quadratic form if $$G (x,y)=\frac 12 (F (x+y)-F (x)-F (y)) $$ is bilinear. In that case $G $ is a symmetric bilinear form and $F (x)=G (x,x) $. Moreover, $F$ is said to be non singular whenever $G (x,y)=0$ for all $y $ implies $x=0$.

  • $\begingroup$ Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch! $\endgroup$ – 20gobbledigook08 Dec 2 '18 at 15:58
  • $\begingroup$ See for example here. $\endgroup$ – Fabio Lucchini Dec 2 '18 at 17:51

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