Bilinear form
on a product of modules
A bilinear mapping , where
is a left unitary
-module,
is a right unitary
-module, and
is a ring with a unit element, which is also regarded as an
-bimodule. If
, one says that
is a bilinear form on the module
, and also that
has a metric structure given by
. Definitions involving bilinear mappings make sense also for bilinear forms. Thus, one speaks of the matrix of a bilinear form with respect to chosen bases in
and
, of the orthogonality of elements and submodules with respect to bilinear forms, of orthogonal direct sums, of non-degeneracy, etc. For instance, if
is a field and
is a finite-dimensional vector space over
with basis
, then for the vectors
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and
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the value of the form will be
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where . The polynomial
in the variables
is sometimes identified with
and is called a bilinear form on
. If the ring
is commutative, a bilinear form is a special case of a sesquilinear form (with the identity automorphism).
Let be a commutative ring. A bilinear form on an
-module
is said to be symmetric (or anti-symmetric or skew-symmetric) if for all
one has
(or
), and is said to be alternating if
. An alternating bilinear form is anti-symmetric; the converse is true only if for any
it follows from
that
. If
has a finite basis, symmetric (or anti-symmetric or alternating) forms on
and only such forms have a symmetric (anti-symmetric, alternating) matrix in this basis. The orthogonality relation with respect to a symmetric or anti-symmetric form on
is symmetric.
A bilinear form on
is said to be isometric with a bilinear form
on
if there exists an isomorphism of
-modules
such that
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for all . This isomorphism is called an isometry of the form and, if
and
, a metric automorphism of the module
(or an automorphism of the form
). The metric automorphisms of a module form a group (the group of automorphisms of the form
); examples of such groups are the orthogonal group or the symplectic group.
Let be a skew-field and let
be a bilinear form on
; let the spaces
and
be finite-dimensional over
; one then has
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and this number is called the rank of . If
is finite-dimensional and
is non-degenerate, then
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and for each basis in
there exists a basis
in
which is dual with respect to
; it is defined by the condition
, where
are the Kronecker symbols. If, in addition,
, then the submodules
and
are said to be the right and the left kernel of
, respectively; for symmetric and anti-symmetric forms the right and left kernels are identical and are simply referred to as the kernel.
Let be a symmetric or an anti-symmetric bilinear form on
. An element
for which
is said to be an isotropic element; a submodule
is said to be isotropic if
, and totally isotropic if
. Totally isotropic submodules play an important role in the study of the structure of bilinear forms (cf. Witt decomposition; Witt theorem; Witt ring). See also Quadratic form for the structure of bilinear forms.
Let be commutative, let
be the
-module of all
-linear mappings from
into
, and let
be the
-module of all bilinear forms on
. For every bilinear form
on
and for each
, the formula
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defines an -linear form on
. Correspondingly, for
the formula
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defines an -linear form on
. The mapping
is an element of
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The mapping in
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is defined in a similar way. The mappings and
define isomorphisms between the
-modules
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and
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A bilinear form is said to be left-non-singular (respectively, right-non-singular) if
(respectively,
) is an isomorphism; if
is both left- and right-non-singular, it is said to be non-singular; otherwise it is said to be singular. A non-degenerate bilinear form may be singular. For free modules
and
of the same finite dimension a bilinear form
on
is non-singular if and only if the determinant of the matrix of
with respect to any bases in
and
is an invertible element of the ring
. The following isomorphisms
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and
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given by a non-singular bilinear form , are defined by the formulas
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and
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The endomorphisms and
are said to be conjugate with respect to the form
if
.
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) |
[2] | S. Lang, "Algebra" , Addison-Wesley (1974) |
[3] | E. Artin, "Geometric algebra" , Interscience (1957) |
Bilinear form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bilinear_form&oldid=30647