bilinear form
Definition.
Let $U,V,W$ be vector spaces^{} over a field $K$. A bilinear map is a function $B:U\times V\to W$ such that

1.
the map $x\mapsto B(x,y)$ from $U$ to $W$ is linear for each $y\in V$

2.
the map $y\mapsto B(x,y)$ from $V$ to $W$ is linear for each $x\in U$.
That is, $B$ is bilinear if it is linear in each parameter, taken separately.
Bilinear forms.
A bilinear form is a bilinear map $B:V\times V\to K$. A $W$valued bilinear form is a bilinear map $B:V\times V\to W$. One often encounters bilinear forms with additional assumptions^{}. A bilinear form is called

•
symmetric^{} if $B(x,y)=B(y,x)$, $x,y\in V$;

•
skewsymmetric if $B(x,y)=B(y,x)$, $x,y\in V$;

•
alternating if $B(x,x)=0$, $x\in V$.
By expanding $B(x+y,x+y)=0$, we can show alternating implies skewsymmetric. Further if $K$ is not of characteristic $2$, then skewsymmetric implies alternating.
Left and Right Maps.
Let $B:U\times V\to W$ be a bilinear map. We may identify $B$ with the linear map ${B}_{\otimes}:U\otimes V\to W$ (see tensor product^{}). We may also identify $B$ with the linear maps
${B}_{L}:U\to L(V,W),{B}_{L}(x)(y)=B(x,y),x\in U,y\in V;$  
${B}_{R}:V\to L(U,W),{B}_{R}(y)(x)=B(x,y),x\in U,y\in V.$ 
called the left and right map, respectively.
Next, suppose that $B:V\times V\to K$ is a bilinear form. Then both ${B}_{L}$ and ${B}_{R}$ are linear maps from $V$ to ${V}^{*}$, the dual vector space of $V$. We can therefore say that $B$ is symmetric if and only if ${B}_{L}={B}_{R}$ and that $B$ is antisymmetric if and only if ${B}_{L}={B}_{R}$. If $V$ is finitedimensional, we can identify $V$ and ${V}^{**}$, and assert that ${B}_{L}={({B}_{R})}^{*}$; the left and right maps are, in fact, dual homomorphisms.
Rank.
Let $B:U\times V\to K$ be a bilinear form, and suppose that $U,V$ are finite dimensional. One can show that $\mathrm{rank}{B}_{L}=\mathrm{rank}{B}_{R}$. We call this integer $\mathrm{rank}B$, the of $B$. Applying the ranknullity theorem^{} to both the left and right maps gives the following results:
$dimU$  $=dim\mathrm{ker}{B}_{L}+\mathrm{rank}B$  
$dimV$  $=dim\mathrm{ker}{B}_{R}+\mathrm{rank}B$ 
We say that $B$ is nondegenerate if both the left and right map are nondegenerate. Note that in for $B$ to be nondegenerate it is necessary that $dimU=dimV$. If this holds, then $B$ is nondegenerate if and only if $\mathrm{rank}B$ is equal to $dimU,dimV$.
Orthogonal complements.
Let $B:V\times V\to K$ be a bilinear form, and let $S\subset V$ be a subspace^{}. The left and right orthogonal complements^{} of $S$ are subspaces ${}^{\u27c2}S,{S}^{\u27c2}\subset V$ defined as follows:
${}^{\u27c2}S=\{u\in V\mid B(u,v)=0\text{for all}v\in S\},$  
${S}^{\u27c2}=\{v\in V\mid B(u,v)=0\text{for all}u\in S\}.$ 
We may also realize ${S}^{\u27c2}$ by considering the linear map ${B}_{R}^{\prime}:V\to {S}^{*}$ obtained as the composition of ${B}_{R}:V\to {V}^{*}$ and the dual homomorphism ${V}^{*}\to {S}^{*}$. Indeed, ${S}^{\u27c2}=\mathrm{ker}{B}_{R}^{\prime}$. An analogous statement can be made for ${}^{\u27c2}S$.
Next, suppose that $B$ is nondegenerate. By the ranknullity theorem we have that
$dimV$  $=dimS+dim{S}^{\u27c2}$  
$=dimS+dim{}^{\u27c2}S.$ 
Therefore, if $B$ is nondegenerate, then
$$dim{S}^{\u27c2}=dim{}^{\u27c2}S.$$ 
Indeed, more can be said if $B$ is either symmetric or skewsymmetric. In this case, we actually have
$${}^{\u27c2}S={S}^{\u27c2}.$$ 
We say that $S\subset V$ is a nondegenerate subspace relative to $B$ if the restriction^{} of $B$ to $S\times S$ is nondegenerate. Thus, $S$ is a nondegenerate subspace if and only if $S\cap {S}^{\u27c2}=\{0\}$, and also $S\cap {}^{\u27c2}S=\{0\}$. Hence, if $B$ is nondegenerate and if $S$ is a nondegenerate subspace, we have
$$V=S\oplus {S}^{\u27c2}=S\oplus {}^{\u27c2}S.$$ 
Finally, note that if $B$ is positivedefinite, then $B$ is necessarily nondegenerate and that every subspace is nondegenerate. In this way we arrive at the following wellknown result: if $V$ is positivedefinite inner product space^{}, then
$$V=S\oplus {S}^{\u27c2}$$ 
for every subspace $S\subset V$.
Adjoints.
Let $B:V\times V\to K$ be a nondegenerate bilinear form, and let $T\in L(V,V)$ be a linear endomorphism^{}. We define the right adjoint ${T}^{\star}\in L(V,V)$ to be the unique linear map such that
$$B(Tu,v)=B(u,{T}^{\star}v),u,v\in V.$$ 
Letting ${T}^{\ast}:{V}^{\ast}\to {V}^{\ast}$ denote the dual homomorphism, we also have
$${T}^{\star}=B_{R}{}^{1}\circ {T}^{\ast}\circ {B}_{R}.$$ 
Similarly, we define the left adjoint ${}^{\star}T\in L(V,V)$ by
$${}^{\star}T=B_{L}{}^{1}\circ {T}^{\ast}\circ {B}_{L}.$$ 
We then have
$$B(u,Tv)=B({}^{\star}Tu,v),u,v\in V.$$ 
If $B$ is either symmetric or skewsymmetric, then ${}^{\star}T={T}^{\star}$, and we simply use ${T}^{\star}$ to refer to the adjoint homomorphism.
Additional remarks.

1.
if $B$ is a symmetric, nondegenerate bilinear form, then the adjoint^{} operation^{} is represented, relative to an orthogonal basis (if one exists), by the matrix transpose.

2.
If $B$ is a symmetric, nondegenerate bilinear form then $T\in L(V,V)$ is then said to be a normal operator (with respect to $B$) if $T$ commutes with its adjoint ${T}^{\star}$.

3.
An $n\times m$ matrix may be regarded as a bilinear form over ${K}^{n}\times {K}^{m}$. Two such matrices, $B$ and $C$, are said to be congruent if there exists an invertible^{} $P$ such that $B={P}^{T}CP$.

4.
The identity matrix^{}, ${I}_{n}$ on ${\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$ gives the standard Euclidean^{} inner product^{} on ${\mathbb{R}}^{n}$.
Title  bilinear form 
Canonical name  BilinearForm 
Date of creation  20130322 12:14:02 
Last modified on  20130322 12:14:02 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  54 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 47A07 
Classification  msc 11E39 
Classification  msc 15A63 
Synonym  bilinear 
Related topic  DualityWithRespectToANonDegenerateBilinearForm 
Related topic  BilinearMap 
Related topic  Multilinear 
Related topic  SkewSymmetricBilinearForm 
Related topic  SymmetricBilinearForm 
Related topic  NonDegenerateBilinearForm 
Defines  rank of bilinear form 
Defines  left map 
Defines  right map 