Symmetric bilinear formFrom Wikipedia, the free encyclopedia

A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms areof great importance in the study of orthogonal polarity and quadrics.

They are also more briefly referred to as just symmetric forms when "bilinear" is understood. They are closelyrelated to quadratic forms; for the details of the distinction between the two, see ε-quadratic forms.

Contents

1 Definition2 Matrix representation3 Orthogonality and singularity4 Orthogonal basis

4.1 Signature and Sylvester's law of inertia4.2 Real case4.3 Complex case

5 Orthogonal polarities6 References

Definition

Let V be a vector space of dimension n over a field K. A map is asymmetric bilinear form on the space if :

The last two axioms only imply linearity in the first argument, but the first immediately implies linearity in thesecond argument then too.

Matrix representation

Let be a basis for V. Define the n×n matrix A by . The matrix A is asymmetric matrix exactly due to symmetry of the bilinear form. If the n×1 matrix x represents a vector v withrespect 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×nmatrix. Now the new matrix representation for the symmetric bilinear form is given by

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Orthogonality and singularity

A symmetric bilinear form is always reflexive. Two vectors v and w are defined to be orthogonal with respect tothe bilinear form B if , which is, due to reflexivity, equivalent with

The radical of a bilinear form B is the set of vectors orthogonal with every other vector in V. One easily checksthat this is a subspace of V. 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 the orthogonal complement is the set of all vectors orthogonal with every vectorin W: it is a subspace of V. When B is non-degenerate, so that the radical of B is trivial, the dimension of =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, there is always an orthogonal basis. This can be proven byinduction.

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 K, the numberof diagonal elements equal to 0, or that are positive or negative, is 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 orthogonalbasis.

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. Zeroeswill appear if and only if the radical is nontrivial.

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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 willappear 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 characteristicdifferent from 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 orthogonalpolarities are induced in this way, and that two symmetric bilinear forms with trivial radical induce the samepolarity 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. GraduateTexts in Mathematics. 136. Springer-Verlag. ISBN 3-540-97839-9. Zbl 0768.00003(http://www.zentralblatt-math.org/zmath/en/search/?q=an:0768.00003&format=complete) .Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrerGrenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016 (http://www.zentralblatt-math.org/zmath/en/search/?q=an:0292.10016&format=complete) .Weisstein, Eric W., "Symmetric Bilinear Form (http://mathworld.wolfram.com/SymmetricBilinearForm.html) " from MathWorld.

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