bilinear form


Let U,V,W be vector spacesMathworldPlanetmath over a field K. A bilinear map is a function B:U×VW such that

  1. 1.

    the map xB(x,y) from U to W is linear for each yV

  2. 2.

    the map yB(x,y) from V to W is linear for each xU.

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×VK. A W-valued bilinear form is a bilinear map B:V×VW. One often encounters bilinear forms with additional assumptionsPlanetmathPlanetmath. A bilinear form is called

By expanding B(x+y,x+y)=0, we can show alternating implies skew-symmetric. Further if K is not of characteristic 2, then skew-symmetric implies alternating.

Left and Right Maps.

Let B:U×VW be a bilinear map. We may identify B with the linear map B:UVW (see tensor productPlanetmathPlanetmathPlanetmath). We may also identify B with the linear maps


called the left and right map, respectively.

Next, suppose that B:V×VK is a bilinear form. Then both BL and BR 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 BL=BR and that B is anti-symmetric if and only if BL=-BR. If V is finite-dimensional, we can identify V and V**, and assert that BL=(BR)*; the left and right maps are, in fact, dual homomorphisms.


Let B:U×VK be a bilinear form, and suppose that U,V are finite dimensional. One can show that rankBL=rankBR. We call this integer rankB, the of B. Applying the rank-nullity theoremMathworldPlanetmath to both the left and right maps gives the following results:

dimU =dimkerBL+rankB
dimV =dimkerBR+rankB

We say that B is non-degenerate if both the left and right map are non-degenerate. Note that in for B to be non-degenerate it is necessary that dimU=dimV. If this holds, then B is non-degenerate if and only if rankB is equal to dimU,dimV.

Orthogonal complements.

Let B:V×VK be a bilinear form, and let SV be a subspaceMathworldPlanetmathPlanetmath. The left and right orthogonal complementsMathworldPlanetmathPlanetmath of S are subspaces S,SV defined as follows:

S={uVB(u,v)=0for all vS},
S={vVB(u,v)=0for all uS}.

We may also realize S by considering the linear map BR:VS* obtained as the composition of BR:VV* and the dual homomorphism V*S*. Indeed, S=kerBR. An analogous statement can be made for S.

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

dimV =dimS+dimS

Therefore, if B is non-degenerate, then


Indeed, more can be said if B is either symmetric or skew-symmetric. In this case, we actually have


We say that SV is a non-degenerate subspace relative to B if the restrictionPlanetmathPlanetmath of B to S×S is non-degenerate. Thus, S is a non-degenerate subspace if and only if SS={0}, and also SS={0}. Hence, if B is non-degenerate and if S is a non-degenerate subspace, we have


Finally, note that if B is positive-definite, then B is necessarily non-degenerate and that every subspace is non-degenerate. In this way we arrive at the following well-known result: if V is positive-definite inner product spaceMathworldPlanetmath, then


for every subspace SV.


Let B:V×VK be a non-degenerate bilinear form, and let TL(V,V) be a linear endomorphismPlanetmathPlanetmath. We define the right adjoint TL(V,V) to be the unique linear map such that


Letting T:VV denote the dual homomorphism, we also have


Similarly, we define the left adjoint TL(V,V) by


We then have


If B is either symmetric or skew-symmetric, then T=T, and we simply use T to refer to the adjoint homomorphism.

Additional remarks.

  1. 1.

    if B is a symmetric, non-degenerate bilinear form, then the adjointPlanetmathPlanetmath operationMathworldPlanetmath is represented, relative to an orthogonal basis (if one exists), by the matrix transpose.

  2. 2.

    If B is a symmetric, non-degenerate bilinear form then TL(V,V) is then said to be a normal operator (with respect to B) if T commutes with its adjoint T.

  3. 3.

    An n×m matrix may be regarded as a bilinear form over Kn×Km. Two such matrices, B and C, are said to be congruent if there exists an invertiblePlanetmathPlanetmathPlanetmathPlanetmath P such that B=PTCP.

  4. 4.

    The identity matrixMathworldPlanetmath, In on n×n gives the standard EuclideanPlanetmathPlanetmathPlanetmath inner productMathworldPlanetmath on n.

Title bilinear form
Canonical name BilinearForm
Date of creation 2013-03-22 12:14:02
Last modified on 2013-03-22 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