"Let Q(v)=<v,v> be the quadratic form associated to a real or hermitian inner product space. ... "

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In my lineair algebra course, it was defined like this (for the real case):

I suppose you know what a billineair map is.
If there is such a billineair map $b:E \times E \to \mathbb{R}$, where E is an n-dimensional Euclidean space, then we can define a map $q:E \to \mathbb{R}$ as $q\left( {\vec x} \right) = b\left( {\vec x,\vec x} \right)$.

We call this q the quadratic form, associated to the billineair map b.

So in general, we have then:

$$q\left( {\vec x} \right) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {a_{ij} x_i x_j } }$$

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