Symmetric bilinear map

Subcase of symmetric multilinear map:

Requires the two inputs $x$ and $y$ to be in the same vector space of course.

The most important example is the dot product, which is also a positive definite symmetric bilinear form.

symmetric bilinear maps that is also a bilinear form.

Like the matrix representation of a bilinear form, it is a matrix, but now the matrix has to be a symmetric matrix.

We can then immediately see that the matrix is symmetric, then so is the form. We have: But because $B(x,y)$ is a scalar, we have: and:

The complex number analogue of a symmetric bilinear form.

The prototypical example of it is the complex dot product.

Note that this form is neither strictly symmetric, it satisfies: where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

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A Hermitian matrix.

Multivariate polynomial where each term has degree 2, e.g.: is a quadratic form because each term has degree 2:but e.g.: is not because the term $x^3$ has degree 3.

More generally for any number of variables it can be written as:

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

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.: But that same matrix could also be written in symmetric form as: so why not I guess, its simpler/more restricted.

Symmetric bilinear form that is also positive definite, i.e.:

A positive definite matrix that is also a symmetric matrix.

Subcase of antisymmetric multilinear map:

Skew-symmetric bilinear map that is also a bilinear form.