Symmetric bilinear form

Inmathematics,asymmetric bilinear formon avector spaceis abilinear mapfrom two copies of the vector space to thefieldofscalarssuch that the order of the two vectors does not affect the value of the map. In other words, it is abilinearfunction$B$that maps every pair$(u,v)$of elements of the vector space$V$to the underlying field such that$B(u,v)=B(v,u)$for every$u$and$v$in$V$.They are also referred to more briefly as justsymmetric formswhen "bilinear" is understood.

Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond tosymmetric matricesgiven abasisforV.Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as anorthogonal basis(at least when thecharacteristicof the field is not 2).

Given a symmetric bilinear formB,the functionq(x) =B(x,x)is the associatedquadratic formon the vector space. Moreover, if the characteristic of the field is not 2,Bis the unique symmetric bilinear form associated withq.

Formal definition

LetVbe a vector space of dimensionnover a fieldK.Amap$B:V\times V\rightarrow K$is a symmetric bilinear form on the space if:

• $B(u,v)=B(v,u)\ \quad \forall u,v\in V$
• $B(u+v,w)=B(u,w)+B(v,w)\ \quad \forall u,v,w\in V$
• $B(\lambda v,w)=\lambda B(v,w)\ \quad \forall \lambda \in K,\forall v,w\in V$

The last two axioms only establish linearity in the first argument, but the first axiom (symmetry) then immediately implies linearity in the second argument as well.

Examples

LetV=Rⁿ,thendimensional real vector space. Then the standarddot productis a symmetric bilinear form,B(x,y) =x⋅y.The matrix corresponding to this bilinear form (see below) on astandard basisis the identity matrix.

LetVbe any vector space (including possibly infinite-dimensional), and assumeTis a linear function fromVto the field. Then the function defined byB(x,y) =T(x)T(y)is a symmetric bilinear form.

LetVbe the vector space of continuous single-variable real functions. For$f,g\in V$one can define$\textstyle B(f,g)=\int _{0}^{1}f(t)g(t)dt$.By the properties ofdefinite integrals,this defines a symmetric bilinear form onV.This is an example of a symmetric bilinear form which is not associated to any symmetric matrix (since the vector space is infinite-dimensional).

Matrix representation

Let$C=\{e_{1},\ldots,e_{n}\}$be a basis forV.Define then×nmatrixAby$A_{ij}=B(e_{i},e_{j})$.The matrixAis asymmetric matrixexactly due to symmetry of the bilinear form. If we let then×1 matrixxrepresent the vectorvwith respect to this basis, and similarly let then×1 matrixyrepresent the vectorw,then$B(v,w)$is given by:

$x^{\mathsf {T}}Ay=y^{\mathsf {T}}Ax.$

SupposeC'is another basis forV,with: ${\begin{bmatrix}e'_{1}&\cdots &e'_{n}\end{bmatrix}}={\begin{bmatrix}e_{1}&\cdots &e_{n}\end{bmatrix}}S$ withSan invertiblen×nmatrix. Now the new matrix representation for the symmetric bilinear form is given by

$A'=S^{\mathsf {T}}AS.$

Orthogonality and singularity

Two vectorsvandware defined to be orthogonal with respect to the bilinear formBifB(v,w) = 0,which, for a symmetric bilinear form, is equivalent toB(w,v) = 0.

Theradicalof a bilinear formBis the set of vectors orthogonal with every vector inV.That this is a subspace ofVfollows from the linearity ofBin each of its arguments. When working with a matrix representationAwith respect to a certain basis,v,represented byx,is in the radical if and only if

$Ax=0\Longleftrightarrow x^{\mathsf {T}}A=0.$

The matrixAis singular if and only if the radical is nontrivial.

IfWis a subset ofV,then itsorthogonal complementW^⊥is the set of all vectors inVthat are orthogonal to every vector inW;it is a subspace ofV.WhenBis non-degenerate, the radical ofBis trivial and the dimension ofW^⊥isdim(W^⊥) = dim(V) − dim(W).

Orthogonal basis

A basis$C=\{e_{1},\ldots,e_{n}\}$is orthogonal with respect toBif and only if:

$B(e_{i},e_{j})=0\ \forall i\neq j.$

When thecharacteristicof the field is not two,Valways has an orthogonal basis. This can be proven byinduction.

A basisCis orthogonal if and only if the matrix representationAis adiagonal matrix.

Signature and Sylvester's law of inertia

In a more general form,Sylvester's law of inertiasays that, when working over anordered field,the numbers of diagonal elements in the diagonalized form of a matrix that are positive, negative and zero respectively are independent of the chosen orthogonal basis. These three numbers form thesignatureof the bilinear form.

Real case

When working in a space over the reals, one can go a bit a further. Let$C=\{e_{1},\ldots,e_{n}\}$be an orthogonal basis.

We define a new basis$C'=\{e'_{1},\ldots,e'_{n}\}$

$e'_{i}={\begin{cases}e_{i}&{\text{if }}B(e_{i},e_{i})=0\\{\frac {e_{i}}{\sqrt {B(e_{i},e_{i})}}}&{\text{if }}B(e_{i},e_{i})>0\\{\frac {e_{i}}{\sqrt {-B(e_{i},e_{i})}}}&{\text{if }}B(e_{i},e_{i})<0\end{cases}}$

Now, the new matrix representationAwill be a diagonal matrix with only 0, 1 and −1 on the diagonal. Zeroes will appear if and only if the radical is nontrivial.

Complex case

When working in a space over the complex numbers, one can go further as well and it is even easier. Let$C=\{e_{1},\ldots,e_{n}\}$be an orthogonal basis.

We define a new basis$C'=\{e'_{1},\ldots,e'_{n}\}$:

$e'_{i}={\begin{cases}e_{i}&{\text{if }}\;B(e_{i},e_{i})=0\\e_{i}/{\sqrt {B(e_{i},e_{i})}}&{\text{if }}\;B(e_{i},e_{i})\neq 0\\\end{cases}}$

Now the new matrix representationAwill be a diagonal matrix with only 0 and 1 on the diagonal. Zeroes will appear if and only if the radical is nontrivial.

Orthogonal polarities

LetBbe a symmetric bilinear form with a trivial radical on the spaceVover the fieldKwithcharacteristicnot 2. One can now define a map from D(V), the set of all subspaces ofV,to itself:

$\alpha:D(V)\rightarrow D(V):W\mapsto W^{\perp }.$

This map is anorthogonal polarityon theprojective spacePG(W). Conversely, one can prove all orthogonal polarities are induced in this way, and that two symmetric bilinear forms with trivial radical induce the same polarity if and only if they are equal up to scalar multiplication.

References

• Adkins, William A.; Weintraub, Steven H. (1992).Algebra: An Approach via Module Theory.Graduate Texts in Mathematics.Vol. 136.Springer-Verlag.ISBN3-540-97839-9.Zbl0768.00003.
• Milnor, J.;Husemoller, D. (1973).Symmetric Bilinear Forms.Ergebnisse der Mathematik und ihrer Grenzgebiete.Vol. 73.Springer-Verlag.ISBN3-540-06009-X.Zbl0292.10016.
• Weisstein, Eric W."Symmetric Bilinear Form".MathWorld.