Multilinear forms on vectors

In general, multilinear mappings from a vector space to scalars are called “forms.” Here we define some forms with regard to the real vector space \({\mathbb{R}^{n}}\); more general definitions may exist in more general settings.

A form is completely symmetric if it is invariant under exchange of any two vector arguments; it is completely anti-symmetric (AKA alternating) if it changes sign under such an exchange. Some classes of forms include:

  • Bilinear form: a bilinear map from two vectors to \({\mathbb{R}}\)
  • Multilinear form: generalizes the above to take any number of vectors
  • Quadratic form: equivalent to a symmetric bilinear form: a quadratic form is a homogeneous polynomial of degree two, i.e. every term has the same number of variables, with no power greater than 2; by considering the variables components of a vector, the polarization identity gives a 1-1 correspondence with symmetric bilinear forms.
  • Algebraic form: generalization of the above to arbitrary degree; equivalent to a completely symmetric multilinear form
  • 2-form: an anti-symmetric bilinear form
  • Exterior form: a completely anti-symmetric multilinear form; exterior forms will be revisited from a different perspective in a subsequent section.

A real inner product is thus a positive definite quadratic form, and is the generalization to arbitrary dimension of the geometrically defined dot product in \({\mathbb{R}^{3}}\). In the context of manifolds, an inner product is called a metric; and in the context of spacetime, variants of the inner product carry specific terminology. We will cover this terminology in the following.

The positive definite definition of a real inner product is sometimes relaxed to only require the product to be nondegenerate (AKA anisotropic), i.e. \({\left\langle v,w\right\rangle =0}\) for all \({w}\) only if \({v=0}\). We will instead refer to this type of form, a nondegenerate symmetric bilinear form, as a pseudo inner product (AKA pseudo-metric); it is characterized by the fact that \({\left\langle v,v\right\rangle }\) can be negative or vanish. A vector \({v}\) is called isotropic (AKA light-like) if \({\left\langle v,v\right\rangle =0}\).

A pseudo inner product does not yield a well-defined norm, but the (not necessarily real) quantity \({\sqrt{\left\langle v,v\right\rangle }}\) is nevertheless sometimes called the “length” of \({v}\). A pseudo inner product also defines orthonormal bases, with the definition modified to allow \({\left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle =\pm\delta^{\mu}{}_{\nu}}\). The number of positive and negative “lengths” of basis vectors turns out to be independent of the choice of basis, and this pair of integers \({\left(r,s\right)}\) is called the signature. We define \({\eta{}_{\mu\nu}\equiv \left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle \equiv\eta^{\mu\nu}}\), so that it is \({\pm1}\) if \({\mu=\nu}\) (with \({r}\) positive values), \({0}\) otherwise. A general signature is called pseudo-Riemannian (AKA pseudo-Euclidean), a signature with \({s=0}\) is called Riemannian (AKA Euclidean), a signature with \({s=1}\) (or sometimes \({r=1}\)) is called Lorentzian (AKA Minkowskian), and the signature (3,1) is called Minkowskian (more specifically this is called the “mostly pluses” signature (AKA relativity, spacelike, or east coast signature), while the signature (1,3), the “mostly minuses” signature (AKA particle physics, timelike, or west coast signature), is also called Minkowskian).

Δ The term “signature” also sometimes refers to the integer \({r-s}\).

Below we summarize the relationships between various forms.
7.forms

The above is an Euler diagram, where the spatial relationships of the boxes indicate their relationships as sets (i.e. intersection, subset, disjoint). We will use these frequently in summarizing the relationships between mathematical concepts.

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