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<Multivariate polynomial> where each term has degree 2, e.g.:
$$
f(x,y) = 2y^2 + 10yx + x^2
$$
is a quadratic form because each term has degree 2:
* $y^2$
* $xy$
* $x^2$
but e.g.:
$$
f(x,y) = 2y^2 + 10yx + x^3
$$
is not because the term $x^3$ has degree 3.

More generally for any number of variables it can be written as:
$$
f(x_1, x_2, \ldots, x_n) = \sum_{i,j} a_i a_j x_i x_j
$$

There is a <1-to-1> relationship between <quadratic forms> and <symmetric bilinear forms>. In matrix representation, this can be written as:
$$
\vec{x}^T B \vec{x}
$$
where $\vec{x}$ contains each of the variabes of the form, e.g. for 2 variables:
$$
\vec{x} = [x, y]
$$

Strictly speaking, the associated <bilinear form> would not need to be a <symmetric bilinear form>, at least for the <real numbers> or <complex numbers> which are <commutative>. E.g.:
$$
\begin{bmatrix}x y\end{bmatrix}
\begin{bmatrix}0 & 1 \\ 2 & 0 \\ \end{bmatrix}
\begin{bmatrix}x \\ y \\ \end{bmatrix}
=
\begin{bmatrix}x y\end{bmatrix}
\begin{bmatrix}y \\ 2x \\\end{bmatrix}
= xy + 2yx
= 3xy
$$
But that same matrix could also be written in symmetric form as:
$$
\begin{bmatrix}0 & 1.5 \\ 1.5 & 0 \\ \end{bmatrix}
$$
so why not I guess, its simpler/more restricted.


Quadratic form

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A <matrix> that equals its <transpose>:
$$
M = M^T
$$

Can represent a <symmetric bilinear form> as shown at <matrix representation of a symmetric bilinear form>, or a <quadratic form>.


Ciro Santilli @cirosantilli 34

 Incoming links: Quadratic form