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I'm having a big problem with what appears to be a basic question concerning quadratic and bilinear forms.

Q: Let $B$ be a symmetric bilinear form with associated quadratic form $Q(x, y, z) = 4xy − 2xz − 4yz − y^2$

Write down the matrix of the bilinear form $B$ with respect to the usual basis. I have the answer here but no idea how it was obtained.

This is my first time asking a question so sorry if the format is incorrect.

Thanks.

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    Please add all relevant details, including the answer, and what you have tried and why you think it is wrong. Looking at the answer won't usually help you *get* the answer. – 2017-01-06

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\begin{align*} \boldsymbol{x}^T \mathbb{B}\, \boldsymbol{x} &= -y^2-4yz-2zx+4xy \\[5pt] \mathbb{B} &= \begin{pmatrix} 0 & 2 & -1 \\ 2 & -1 & -2 \\ -1 & -2 & 0 \end{pmatrix} \end{align*}

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    Thanks for the answer. Could you tell me how you obtained the matrix for B? – 2017-01-06
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    In general,$$\begin{pmatrix} x & y & z \\ \end{pmatrix}\begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix}=ax^2+by^2+cz^2+2fyz+2gzx+2hxy$$ and do it backward. Are you OK in finding the eigenvectors or even the orthogonal bases? – 2017-01-06
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    Once I knew the matrix B finding the eigenvectors and orthonormal bases were no problem. I just had initial problems finding B. But this helped a lot, thanks for your answer! – 2017-01-06