Symmetric bilinear forms, quadratic forms and matrices

Question

I have computed B=$ \left( \begin{array}{ccc} 0 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 1 \end{array} \right) $

Is this correct? If so, even though I may have achieved the correct answer, can you explain the general method for doing this?

Answer 1

You have to divide by two (when in characteristic$\;\neq2\;$) each coefficient of the mixed variables $\;xy\,,\,\,xz\,,\,\,yz\;$ , but you multiplied them by two! It should be

$$\begin{pmatrix}0&1&-1\\ 1&2&0\\-1&0&1\end{pmatrix}$$

Answer 2

A quadratic form on $K^3$ is a polynomial $Q\in K[x,y,z]$ of the form $$ Q(x,y,z)=a\, x^2+b\,y^2+c\,z^2+d\,xy+e\,xz+f\,yz $$ where $a,b,c,d,e,f\in K$. The matrix representing $Q$ is the matrix $B$ satisfying $$ Q(x,y,z)= \begin{bmatrix} x&y&z \end{bmatrix} B \begin{bmatrix} x\\y\\z \end{bmatrix} $$ Can you unravel these equations and check your answer?