PlanetMath: bilinear form

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bilinear form

(Definition)

Definition.

Let

be vector spaces over a field

. A bilinear map is a function $B: U \times V \to W$ such that

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

That is,

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

-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 , $x,y \in V$ ;
skew-symmetric if , $x,y \in V$ ;
alternating if , $x \in V$ .

By expanding

, we can show alternating implies skew-symmetric. Further if

is not of characteristic

, then skew-symmetric implies alternating.

Left and Right Maps.

Let $B:U\times V\to W$ be a bilinear map. We may identify

with the linear map $B_\otimes: U\otimes V\to W$ (see tensor product). We may also identify

with the linear maps

	$\displaystyle B_L:U\to L(V,W),\qquad B_L(x)(y) = B(x,y),\; x\in U,\; y\in V;$
	$\displaystyle B_R:V\to L(U,W),\qquad B_R(y)(x) = B(x,y),\; x\in U,\; y\in V.$

called the left and right map, respectively.

Next, suppose that $B:V\times V\to K$ is a bilinear form. Then both and are linear maps from to , the dual vector space of . We can therefore say that is symmetric if and only if and that is anti-symmetric if and only if . If is finite-dimensional, we can identify and $V^{**}$ , and assert that ; the left and right maps are, in fact, dual homomorphisms.

Rank.

Let $B:U\times V\to W$ be a bilinear map, and suppose that

are finite dimensional. One can show that $\operatorname{rank}B_L = \operatorname{rank}B_R$ . We call this integer $\operatorname{rank}B$ , the rank of

. Applying the rank-nullity theorem to both the left and right maps gives the following results:

$\displaystyle \dim U$	$\displaystyle = \dim \ker {B_L} + \operatorname{rank}B$
$\displaystyle \dim V$	$\displaystyle = \dim \ker {B_R} + \operatorname{rank}B$

We say that

is non-degenerate if both the left and right map are non-degenerate. Note that in order for

to be non-degenerate it is necessary that $\dim U = \dim V$ . If this holds, then

is non-degenerate if and only if $\operatorname{rank}B$ is equal to $\dim U, \dim V$ .

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

are subspaces ${^\perp}S, S^\perp \subset V$ defined as follows:

	$\displaystyle {}^\perp S = \{ u\in V \mid B(u,v) = 0 \;$ for all $\displaystyle v \in S \},$
	$\displaystyle S^\perp = \{ v\in V \mid B(u,v) = 0 \;$ for all $\displaystyle u \in S \} .$

We may also realize $S^\perp$ by considering the linear map $B_{R}': V\to S^*$ obtained as the composition of $B_R: V\to V^*$ and the dual homomorphism $V^*\to S^*$ . Indeed, $S^\perp = \ker B^\prime_R$ . An analogous statement can be made for ${}^\perp S$ .

Next, suppose that is non-degenerate. By the rank-nullity theorem we have that

$\displaystyle \dim V$	$\displaystyle = \dim S + \dim S^\perp$
	$\displaystyle = \dim S + \dim {}^\perp S.$

Therefore, if

is non-degenerate, then

$\displaystyle \dim S^\perp = \dim {}^\perp S.$

Indeed, more can be said if

is either symmetric or skew-symmetric. In this case, we actually have

$\displaystyle {}^\perp S = S^\perp.$

We say that $S\subset V$ is a non-degenerate subspace relative to if the restriction of to $S\times S$ is non-degenerate. Thus, is a non-degenerate subspace if and only if $S \cap S^\perp = \{0\}$ , and also $S\cap {}^\perp S = \{ 0\}$ . Hence, if is non-degenerate and if is a non-degenerate subspace, we have

$\displaystyle V = S \oplus S^\perp = S\oplus {}^\perp S.$

Finally, note that if

is positive-definite, then

is necessarily non-degenerate and that every subspace is non-degenerate. In this way we arrive at the following well-known result: if

is positive-definite inner product space, then

$\displaystyle V = S \oplus S^\perp$

for every subspace $S\subset V$ .

Adjoints.

Let $B: V \times V \rightarrow K$ be a non-degenerate 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

$\displaystyle B(Tu, v) = B(u, T^\star v) ,\quad u,v \in V.$

Letting $T^\ast : V^\ast \to V^\ast$ denote the dual homomorphism, we also have

$\displaystyle T^\star = {B_R}^{-1} \circ T^\ast \circ B_R.$

Similarly, we define the left adjoint ${}^\star T\in L(V,V)$ by

$\displaystyle {}^\star T = {B_L}^{-1} \circ T^\ast \circ B_L.$

We then have

$\displaystyle B(u, Tv) = B({}^\star T u, v) ,\quad u,v \in V.$

is either symmetric or skew-symmetric, then ${}^\star T = T^\star$ , and we simply use $T^\star$ to refer to the adjoint homomorphism.

Additional remarks.

if is a symmetric, non-degenerate bilinear form, then the adjoint operation is represented, relative to an orthogonal basis (if one exists), by the matrix transpose.
If is a symmetric, non-degenerate bilinear form then $T\in L(V,V)$ is then said to be a normal operator (with respect to ) if commutes with its adjoint $T^\star$ .
An $n \times m$ matrix may be regarded as a bilinear form over $K^n\times K^m$ . Two such matrices, and , are said to be congruent if there exists an invertible such that $B = P^{T}CP$ .
The identity matrix, on $\mathbb{R}^n\times \mathbb{R}^n$ gives the standard Euclidean inner product on $\mathbb{R}^n$ .

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"bilinear form" is owned by rmilson. [ full author list (5) | owner history (2) ]

(view preamble)

Other names:

bilinear

Also defines:

rank of bilinear form, left map, right map

Attachments:: matrix representation of a bilinear form (Definition) by vitriol
canonical basis for symmetric bilinear forms (Definition) by Mathprof

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Cross-references: inner product, Euclidean, identity matrix, invertible, congruent, normal operator, transpose, matrix, basis, orthogonal, operation, adjoint, right adjoint, endomorphism, inner product space, restriction, composition, orthogonal complements, right, subspace, necessary, non-degenerate, rank-nullity theorem, integer, finite dimensional, dual homomorphisms, finite-dimensional, anti-symmetric, tensor product, linear map, characteristic, implies, alternating, skew-symmetric, symmetric, parameter, function, bilinear map, field, vector spaces
There are 49 references to this entry.

This is version 47 of bilinear form, born on 2002-01-24, modified 2006-11-06.
Object id is 1612, canonical name is BilinearForm.
Accessed 22820 times total.

Classification:

AMS MSC:	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)
	47A07 (Operator theory :: General theory of linear operators :: Forms )

Pending Errata and Addenda

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