In
mathematics, specifically
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
, a degenerate bilinear form on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' is a
bilinear form such that the map from ''V'' to ''V''
∗ (the
dual space of ''V'' ) given by is not an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. An equivalent definition when ''V'' is
finite-dimensional is that it has a non-trivial kernel: there exist some non-zero ''x'' in ''V'' such that
:
for all
Nondegenerate forms
A nondegenerate or nonsingular form is a
bilinear form that is not degenerate, meaning that
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, or equivalently in finite dimensions,
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
:
for all
implies that
.
The most important examples of nondegenerate forms are
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s and
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
s.
Symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map
be an isomorphism, not positivity. For example, a
manifold with an inner product structure on its
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s is a
Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
.
Using the determinant
If ''V'' is finite-dimensional then, relative to some
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for ''V'', a bilinear form is degenerate if and only if the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the associated
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is zero – if and only if the matrix is ''
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
'', and accordingly degenerate forms are also called singular forms. Likewise, a nondegenerate form is one for which the associated matrix is
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
, and accordingly nondegenerate forms are also referred to as non-singular forms. These statements are independent of the chosen basis.
Related notions
If for a
quadratic form ''Q'' there is a non-zero vector ''v'' ∈ ''V'' such that ''Q''(''v'') = 0, then ''Q'' is an
isotropic quadratic form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...
. If ''Q'' has the same sign for all non-zero vectors, it is a
definite quadratic form
In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical de ...
or an anisotropic quadratic form.
There is the closely related notion of a
unimodular form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linear ...
and a
perfect pairing
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
; these agree over
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
but not over general
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
Examples
The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map
be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a pseudo-Riemannian manifold.
Infinite dimensions
Note that in an infinite-dimensional space, we can have a bilinear form ƒ for which
is
injective but not
surjective. For example, on the space of
continuous functions on a closed bounded
interval, the form
:
is not surjective: for instance, the
Dirac delta functional
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution (mathematics), distribution over the real numbers, whose value is zero everywhere except at zero, and who ...
is in the dual space but not of the required form. On the other hand, this bilinear form satisfies
:
for all
implies that
In such a case where ƒ satisfies injectivity (but not necessarily surjectivity), ƒ is said to be ''weakly nondegenerate''.
Terminology
If ''f'' vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form ''f'' on ''V'' the set of vectors
:
forms a totally degenerate
subspace of ''V''. The map ''f'' is nondegenerate if and only if this subspace is trivial.
Geometrically, an
isotropic line
In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, an ...
of the quadratic form corresponds to a point of the associated
quadric hypersurface in
projective space. Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a
singularity. Hence, over an
algebraically closed field,
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
guarantees that the quadratic form always has isotropic lines, while the bilinear form has them if and only if the surface is singular.
See also
*
*
Citations
{{Topological vector spaces
Bilinear forms
Functional analysis
pl:Forma dwuliniowa#Własności