Ciro Santilli @cirosantilli 34

 Incoming links: Symmetric bilinear form

Bilinear map  Updated 2024-12-15  +Created 1970-01-01

Linear map of two variables.

More formally, given 3 vector spaces X, Y, Z over a single field, a bilinear map is a function from:

f : X \times Y \to Z

that is linear on the first two arguments from X and Y, i.e.:

f (a_{1} x_{1} + a_{2} x_{2}, y) = a_{1} f (x_{1}, y) + a_{2} f (x_{2}, y)

Note that the definition only makes sense if all three vector spaces are over the same field, because linearity can mix up each of them.

The most important example by far is the dot product from

R^{n} \times R^{n} \to R

, which is more specifically also a symmetric bilinear form.

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Hermitian form  Updated 2024-12-15  +Created 1970-01-01

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:

< x, y > = \overline{< y, x >}

where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

mathworld.wolfram.com/HermitianForm.html

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Minkowski inner product  Updated 2024-12-15  +Created 1970-01-01

This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.

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Minkowski inner product matrix  Updated 2024-12-15  +Created 1970-01-01

The matrix representation of the symmetric bilinear form that is the Minkowski inner product.

Since that is a symmetric bilinear form, the associated matrix is a symmetric matrix.

By default, we will use the time negative representation unless stated otherwise:

η_{μ ν} = ⎣ ⎢ ⎢ ⎢ ⎡ - 1 000 010000100001 ⎦ ⎥ ⎥ ⎥ ⎤

but another equivalent one is to use a time positive representation:

η_{μ ν} = ⎣ ⎢ ⎢ ⎢ ⎡ 1000 0 - 1 00 00 - 1 0 000 - 1 ⎦ ⎥ ⎥ ⎥ ⎤

The matrix is typically denoted by the Greek letter eta.

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Positive definite symmetric bilinear form  Updated 2024-12-15  +Created 1970-01-01

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

\forall x, B (x, x) > 0

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Quadratic form  Updated 2024-12-15  +Created 1970-01-01

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:

$y^{2}$
$x y$
$x^{2}$

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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Symmetric matrix  Updated 2024-12-15  +Created 1970-01-01

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.

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Symplectic group  Updated 2024-12-15  +Created 1970-01-01

Intuition, please? Example? mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.

Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has:

O^{T} I O = I

and its generalization the indefinite orthogonal group has:

O^{T} S O = I

where S is symmetric. So for the symplectic group we have matrices Y such as:

Y^{T} A Y = I

where A is antisymmetric. This is explained at: www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.

Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.

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