TheInfoList

In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, 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 matrix (math ...
, a degenerate bilinear form on a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''V'' is a
bilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
such that the map from ''V'' to ''V'' (the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by consta ...
of ''V'') given by is not an
isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... . An equivalent definition when ''V'' is finite-dimensional is that it has a non-trivial
kernel Kernel may refer to: Computing * Kernel (operating system) The kernel is a computer program at the core of a computer's operating system that has complete control over everything in the system. It is the "portion of the operating system co ...
: there exist some non-zero ''x'' in ''V'' such that :$f\left(x,y\right)=0\,$ for all $y \in V.$

Nondegenerate forms

A nondegenerate or nonsingular form is a
bilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
that is not degenerate, meaning that $v \mapsto \left(x \mapsto f\left(x,v\right)\right)$ is an
isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... , or equivalently in finite dimensions, if and only if :$f\left(x,y\right)=0\,$ for all $y \in V \setminus \$ implies that $x = 0$. The most important examples of nondegenerate forms are
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
s and
symplectic formIn mathematics, a symplectic vector space is a vector space ''V'' over a Field (mathematics), field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping that is ...
s. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map $V \to V^*$ be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a
Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
, while relaxing this to a symmetric nondegenerate form yields a
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
.

Using the determinant

If ''V'' is
finite-dimensional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
then, relative to some
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...
for ''V'', a bilinear form is degenerate if and only if the
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... of the associated
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
is zero – if and only if the matrix is ''singular,'' 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 Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic tech ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
''Q'' there is a 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 ...
. If ''Q'' has the same sign for all vectors, it is a
definite quadratic formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
or an anisotropic quadratic form. There is the closely related notion of a
unimodular form In mathematics, a bilinear form on a vector space ''V'' is a bilinear map , where ''K'' is the field (mathematics), field of scalar (mathematics), scalars. In other words, a bilinear form is a function that is linear map, linear in each argument s ...
and a
perfect pairing In mathematics, a bilinear form on a vector space ''V'' is a bilinear map , where ''K'' is the field (mathematics), field of scalar (mathematics), scalars. In other words, a bilinear form is a function that is linear map, linear in each argument s ...
; these agree over fields but not over general rings.

Examples

The most important examples of nondegenerate forms are
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
s and
symplectic formIn mathematics, a symplectic vector space is a vector space ''V'' over a Field (mathematics), field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping that is ...
s. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map $V \to V^*$ be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a
Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
, while relaxing this to a symmetric nondegenerate form yields a
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
.

Infinite dimensions

Note that in an infinite dimensional space, we can have a bilinear form ƒ for which $v \mapsto \left(x \mapsto f\left(x,v\right)\right)$ is
injective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
but not
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. For example, on the space of
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...
s on a closed bounded interval, the form :$f\left(\phi,\psi\right) = \int\psi\left(x\right)\phi\left(x\right) dx$ is not surjective: for instance, the
Dirac delta functional Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. Dirac made fundamental contributions to the early developmen ...
is in the dual space but not of the required form. On the other hand, this bilinear form satisfies :$f\left(\phi,\psi\right)=0\,$ for all $\,\phi$ implies that $\psi=0.\,$ In such a case where ƒ satisfies injectivity (but not necessarily surjectivity), ƒ is said to be ''weakly nondegenerate''.

Terminology

If ƒ vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form ƒ on ''V'' the set of vectors :$\$ forms a totally degenerate subspace of ''V''. The map ƒ is nondegenerate
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
this subspace is trivial. Geometrically, an
isotropic line In the geometry of quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
of the quadratic form corresponds to a point of the associated quadric hypersurface in
projective space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a
singularity Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiabl ...
. Hence, over an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
Hilbert's nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''wikt:Satz, Satz'') is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of alg ... guarantees that the quadratic form always has isotropic lines, while the bilinear form has them if and only if the surface is singular.