# More questions on quadratic forms over field

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!

• The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms. – amd Dec 2 '18 at 3:18
• 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 $B(x,y) = x^\top \frac{A+A^\top}{2} y, q(x) = x^\top \frac{A+A^\top}{2} x$ – reuns Dec 2 '18 at 4:01
• @amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then? – 20gobbledigook08 Dec 2 '18 at 13:41
• @reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though! – 20gobbledigook08 Dec 2 '18 at 13:50

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 $$x=0$$.

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