Quadratic form

Multivariate polynomial where each term has degree 2, e.g.:

f (x, y) = 2 y^{2} + 10 y x + x^{2}

is a quadratic form because each term has degree 2:

but e.g.:

f (x, y) = 2 y^{2} + 10 y x + 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}, \dots, 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:

x^{T} B x

where

x

contains each of the variabes of the form, e.g. for 2 variables:

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.:

[x y] [0210] [x y] = [x y] [y 2 x] = x y + 2 y x = 3 x y

But that same matrix could also be written in symmetric form as:

[0 1.5 1.5 0]

so why not I guess, its simpler/more restricted.

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