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Consider a quadratic form $Q = a_{ij}x_{i}x_{j}$, where the summation from 1 to $n$, the number of independent variables is implied on $i$ and $j$.

By this definition $a_{ij}$ is not symmetric, but it seems to me that most quadratic forms have $a_{ij}$ symmetric. Is this just a coincidence, or is it true that for all quadratic forms $a_{ij}$ is symmetric?

We speak of $Q$ as being a quadratic form over the field $\mathbb{K}$, so that $a_{ij} \in \mathbb{K}$. But do we simply always assume that the coordinates $x_i$ are real? Shouldn't this be specified?

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  • $\begingroup$ any particular book you are reading? $\endgroup$
    – Will Jagy
    Sep 2, 2015 at 1:59

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A quadratic form is, by definition, a function $x \mapsto f(x,x)$, where $f$ is a symmetric bilinear form on the vector space you're working with over the field that you specify. The coefficients $a_{ij}$ correspond to the entries of the representative matrix of the quadratic form, which is also the representative matrix of the symmetric bilinear form which induces our quadratic form. The representative matrix of a symmetric bilinear form is a symmetric matrix.

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    $\begingroup$ $f$ could be any bilinear form, but it turns out that any such $f$ can be decomposed into a symmetric and an alternating part in a unique way, and the associated quadratic form depends only on the symmetric part. I guess this is why they usually only define a quadratic form associated to a symmetric bilinear form, but I think this point should be emphasized more than it is. $\endgroup$
    – Steven Gubkin
    Sep 2, 2015 at 2:08

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