Symmetrical bilinear form
A symmetrical bilinear form is the name given to a bilinear Forme on a vector Space which is symmetrical. The symmetrical bilinear forms play a big role in the study of the quadric .
DefinitionThat is to say a vector space of dimension on a commutative body . A application is a symmetrical bilinear form on space if:
Remarque: The last two axioms imply only the linearity compared to the “first variable” but the first makes it possible to deduce from it the linearity compared to the “second variable”.
That is to say a base of a vector space . Let us define the matrix order by . The matrix is symmetrical according to the symmetry of the bilinear form. If the matrix of the type represents the coordinates of a vector compared to this base, and in a similar way represents the coordinates of a vector , then is equal to:
Let us suppose that is another base of , consider the Matrice of passage (invertible) of order of the base at the base . Maintaining in this new base the matric representation of the symmetrical bilinear form is given by
Orthogonality and singularity
A symmetrical bilinear form is always reflexive. By definition, two vectors and are orthogonal for the bilinear form if , which, thanks to reflexivity, is equivalent to .
The core of a bilinear form is the whole of the orthogonal vectors to any other vector of . It is easy to check that it is a subspace of . When we work with a matric representation of relative at a certain base, a vector represented by its matrix column of the coordinates , belongs to the core if and only if
The matrix is noninvertible or singular if and only if the core of is nonreduced to the singleton null vector, i.e. noncommonplace.
If is a vectorial subspace of , then , the whole of all the orthogonal vectors to any vector of is also a subspace of . When the core of is commonplace, the dimension of .
Orthogonal basesA base is othogonale for if:
When the characteristic of the body is different from two, there exists always an orthogonal base. That can be shown by recurrence.
A base is othogonale if and only if the matrix representative in this base is a diagonal Matrice.
Signature and law of inertia of SylvesterIn its most general form, the Loi of inertia of Sylvester affirms, that while working on a Corps ordered , the number of diagonal elements null, or strictly positive, or strictly negative, is independent of the selected orthogonal base. These three numbers constitute the signature bilinear form.
Real caseWhile working on the body of realities, it is possible to go a little further. That is to say an orthogonal base.
Let us define new bases by
Now, the matrix representing the symmetrical bilinear form, in this new base, is a diagonal matrix having of the 0 or the 1 only on its diagonal. From the zeros appear on the diagonal if and only if the core is noncommonplace.
While working on the body of the complex numbers, one can establish a result similar to that of the real case.
That is to say an orthogonal base.
For all such as , let us note one of the square roots of .
Let us define a new base by
Now, the matrix in the new base is a diagonal matrix having only of the 0 or 1 on the diagonal. From the zeros appear if and only if the core is noncommonplace.
- bilinear Form
- square Form
- Law of inertia of Sylvester
- traditional quadratic Forms and groups of Rene Deheuvels. Editions puf.
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