Symmetric bilinear form
A symmetric bilinear form on a vector space is a linear map from two copies of the vector space to the field of scalars in such a way that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also more briefly referred to as just symmetric forms when "bilinear" is understood.
Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices. 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 an orthogonal basis (at least when the characteristic of the field is not 2).
Given a symmetric bilinear form B, the function q(x):=B(x,x) defines a quadratic form on the vector space. Moreover, if the characteristic of the field is not 2, all quadratic forms are made in this way.
Definition and examples
Let V be a vector space of dimension n over a field K. A map is a symmetric bilinear form on the space if:
The last two axioms only imply linearity in the first argument, but the first axiom then immediately implies linearity in the second argument as well.
First examples
- Let V=Rn, the n dimensional real vector space. Then the standard dot product is a symmetric bilinear form, B(x,y):=x · y. The matrix corresponding to this bilinear form (see below) is the identity matrix.
- Let V be any vector space (including possibly infinite-dimensional), and assume T1 and T2 are linear functions from V to the field. Then the function defined by B(x,y):=T1(x)T2(x) is a symmetric bilinear form. This demonstrates the existence of symmetric bilinear forms which do not come from symmetric matrices.
- Let V be the vector space of continuous single-variable real functions. For one can define By the properties of definite integrals, this defines a symmetric bilinear form on V.
Matrix representation
Let be a basis for V. Define the n×n matrix A by . The matrix A is a symmetric matrix exactly due to symmetry of the bilinear form. If the n×1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then is given by :
Suppose C' is another basis for V, with : with S an invertible n×n matrix. Now the new matrix representation for the symmetric bilinear form is given by
Orthogonality and singularity
A symmetric bilinear form is always reflexive. Two vectors v and w are defined to be orthogonal with respect to the bilinear form B if B(v, w) = 0, which is, due to reflexivity, equivalent to B(w, v) = 0.
The radical of a bilinear form B is the set of vectors orthogonal with every vector in V. That this is a subspace of V follows from the linearity of B in each of its arguments. When working with a matrix representation A with respect to a certain basis, v, represented by x, is in the radical if and only if
The matrix A is singular if and only if the radical is nontrivial.
If W is a subset of V, then its orthogonal complement W⊥ is the set of all vectors in V that are orthogonal to every vector in W; it is a subspace of V. When B is non-degenerate, the radical of B is trivial and the dimension of W⊥ is dim(W⊥) = dim(V) − dim(W).
Orthogonal basis
A basis is orthogonal with respect to B if and only if :
When the characteristic of the field is not two, V always has an orthogonal basis. This can be proven by induction.
A basis C is orthogonal if and only if the matrix representation A is a diagonal matrix.
Signature and Sylvester's law of inertia
In its most general form, Sylvester's law of inertia says that, when working over an ordered field, the numbers of diagonal elements which are positive, zero and negative respectively are independent of the chosen orthogonal basis. These three numbers form the signature of the bilinear form.
Real case
When working in a space over the reals, one can go a bit a further. Let be an orthogonal basis.
We define a new basis
Now, the new matrix representation A will 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 be an orthogonal basis.
We define a new basis :
Now the new matrix representation A will 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
Let B be a symmetric bilinear form with a trivial radical on the space V over the field K with characteristic not 2. One can now define a map from D(V), the set of all subspaces of V, to itself:
This map is an orthogonal polarity on the projective space PG(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. ISBN 3-540-97839-9. Zbl 0768.00003.
- Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
- Weisstein, Eric W. "Symmetric Bilinear Form". MathWorld.