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?