Bilinear form

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

on a product of modules $V\times W$

A bilinear mapping $f: V\times W\to A$, where $V$ is a left unitary $A$-module, $W$ is a right unitary $A$-module, and $A$ is a ring with a unit element, which is also regarded as an $(A<A)$-bimodule. If $V=W$, one says that $f$ is a bilinear form on the module $V$, and also that $V$ has a metric structure given by $f$. 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 $V$ and $W$, of the orthogonality of elements and submodules with respect to bilinear forms, of orthogonal direct sums, of non-degeneracy, etc. For instance, if $A$ is a field and $V=W$ is a finite-dimensional vector space over $A$ with basis $e_1,\dots,e_n$, then for the vectors

$$v = v_1e_1+\cdots+v_ne_n$$ and

$$w = w_1e_1+\cdots+w_ne_n$$ the value of the form will be

$$f(v,w)=\sum_{i,j=1}^n a_{ij}v_iw_j,$$ where $a_{ij} = f(e_i,e_j)$. The polynomial $\sum_{i,j=1}^n a_{ij}v_iw_j$ in the variables $v_1,\dots,v_n,w_1,\dots,w_n$ is sometimes identified with $f$ and is called a bilinear form on $V$. If the ring $A$ is commutative, a bilinear form is a special case of a sesquilinear form (with the identity automorphism).

Let $A$ be a commutative ring. A bilinear form on an $A$-module $V$ is said to be symmetric (or anti-symmetric or skew-symmetric) if for all $v_1,v_2\in V$ one has $f(v_1,v_2) = f(v_2,v_1)$ (or $f(v_1,v_2) = -f(v_2,v_1)$), and is said to be alternating if $f(v,v)=0$. An alternating bilinear form is anti-symmetric; the converse is true only if for any $a\in A$ it follows from $2a=0$ that $a=0$. If $V$ has a finite basis, symmetric (or anti-symmetric or alternating) forms on $V$ 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 $V$ is symmetric.

A bilinear form $f$ on $V$ is said to be isometric with a bilinear form $g$ on $W$ if there exists an isomorphism of $A$-modules $\def\phi{\varphi}\phi:V\to W$ such that

$$g(\phi(v),\phi(w)) = f(v,w)$$ for all $v\in V$. This isomorphism is called an isometry of the form and, if $V=W$ and $f=g$, a metric automorphism of the module $V$ (or an automorphism of the form $f$). The metric automorphisms of a module form a group (the group of automorphisms of the form $f$); examples of such groups are the orthogonal group or the symplectic group.

Let $A$ be a skew-field and let $f$ be a bilinear form on $V\times W$; let the spaces $V/W^\perp$ and $W/V^\perp$ be finite-dimensional over $A$; one then has

$$\dim V/W^\perp = \dim W/V^\perp$$ and this number is called the rank of $f$. If $V$ is finite-dimensional and $f$ is non-degenerate, then

$$\dim V = \dim W$$ and for each basis $v_1,\dots,v_n$ in $V$ there exists a basis $w_1,\dots,w_n$ in $W$ which is dual with respect to $f$; it is defined by the condition $f(v_i,w_j)=\delta_{ij}$, where $\delta{ij}$ are the Kronecker symbols. If, in addition, $V=W$, then the submodules $V^\perp$ and $W^\perp$ are said to be the right and the left kernel of $f$, respectively; for symmetric and anti-symmetric forms the right and left kernels are identical and are simply referred to as the kernel.

Let $f$ be a symmetric or an anti-symmetric bilinear form on $V$. An element $v\in V$ for which $f(v,v)=0$ is said to be an isotropic element; a submodule $M\subset V$ is said to be isotropic if $M\cap M^\perp \ne \{0\}$, and totally isotropic if $M\subset M^\perp$. 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 $A$ be commutative, let $\def\Hom{ {\rm Hom}}\Hom_A(V,W)$ be the $A$-module of all $A$-linear mappings from $V$ into $W$, and let $L_2(V,W)$ be the $A$-module of all bilinear forms on $V\times W$. For every bilinear form $f$ on $V\times W$ and for each $v_0\in V$, the formula

$$l_{f,v_0}(w) = f(v_0,w),\quad w\in W,$$ defines an $A$-linear form on $W$. Correspondingly, for $w_0\in W$ the formula

$$r_{f,w_0}(v) = f(v,w_0),\quad v\in V,$$ defines an $A$-linear form on $V$. The mapping $l_f:v_0\mapsto l_{f,v_0}$ is an element of

$$\Hom_A(V,\Hom_A(W,A)).$$ The mapping $r_f$ in

$$\Hom_A(W,\Hom_A(V,A)).$$ is defined in a similar way. The mappings $f\mapsto l_f$ and $f\mapsto r_f$ define isomorphisms between the $A$-modules

$$L_2(V,W,A) \to \Hom_A(V,\Hom_A(W,A)$$ and

$$L_2(V,W,A) \to \Hom_A(W,\Hom_A(V,A)$$ A bilinear form $f$ is said to be left-non-singular (respectively, right-non-singular) if $l_f$ (respectively, $r_f$) is an isomorphism; if $f$ 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 $V$ and $W$ of the same finite dimension a bilinear form $f$ on $V\times W$ is non-singular if and only if the determinant of the matrix of $f$ with respect to any bases in $V$ and $W$ is an invertible element of the ring $A$. The following isomorphisms

$$\Hom_A(V,V) \stackrel{l}{\to}L_2(V,W,A)$$ and

$$\Hom_A(W,W) \stackrel{r}{\to}L_2(V,W,A),$$ given by a non-singular bilinear form $f$, are defined by the formulas

$$l(\phi)(v,w) = f(\phi(v),w)$$ and

$$r(\psi)(v,w) = f(v,\psi(w)).$$ The endomorphisms $\phi\in \Hom_A(V,V)$ and $\psi\in \Hom_A(W,W)$ are said to be conjugate with respect to the form $f$ if $\psi = (r^{-1}\circ l)(\phi)$.

References

[Ar]	E. Artin, "Geometric algebra", Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[Bo]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra", 1, Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207
[La]	S. Lang, "Algebra", Addison-Wesley (1974) MR0277543 Zbl 0984.00001