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Action of on the space of quadratic forms on

Let be a symmetric matrix and its associated quadratic form. If is an invertible matrix, then . Notice that the function is a new quadratic form.

The aim of the following exercises is to study the induced action of the general linear group on the space of quadratic forms.

Exercise 3.1 (05)   Show that the map defines a right action of on the (vector) space of symmetric matrices.

Since this action is not transitive, we are interested in studying its orbits. In other words, given two symmetric matrices and , we wish to determine whether there exists an invertible matrix such that .

Definition 3.1   Let be a quadratic form on . The nullity of is the nullity of its associated symmetric bilinear form. The index of is if there exists an -dimensional subspace such that the restriction of to is negative definite, and there exist no higher dimensional subspaces with this property.

Exercise 3.2 (05)   Let be a quadratic form on , let be an invertible matrix, and let be the quadratic form . Show that the nullity and the index of and are equal.

Exercise 3.3 (05)   Show that if is a diagonal matrix, then the nullity of the associated quadratic form equals the number of zeros on the diagonal and the index equals the number of negative entries.

The main result in this section is the following theorem:

Theorem 3.1   Let and be quadratic forms on . If the nullity and the index of and are equal, then there exists an invertible matrix such that for all .

In particular, the set of orbits of the action of the general linear group on the space of quadratic forms on is in one-to-one correspondence with the set of pairs , where and are nonnegative integers with .

The proof of theorem 3.1 is quite simple if you remember something of your course on linear algebra.

Exercise 3.4 (-10)   Review your notes on linear algebra and relearn the theorem (and its proof) that states that if is an symmetric matrix, then there exists an orthogonal matrix such that is diagonal.

Notice that if is orthogonal, . This means that every orbit contains a diagonal matrix.

Exercise 3.5 (05)   Show that if is a diagonal matrix, there exists an invertible matrix such that is diagonal and all its entries are either 1, -1, or 0. Hint: can be chosen to be a diagonal matrix.

Exercise 3.6 (05)   Let be an diagonal matrix such that entries in the diagonal are equal to zero, entries are equal to -1, and the rest are equal to 1. Show that there exists an invertible matrix such that is a diagonal matrix whose first diagonal entries equal to 1, the following entries equal to -1, and the following entries equal to zero. Hint: take to be a permutation matrix.

Exercise 3.7 (05)   Show that if is an symmetric matrix whose associated quadratic form has nullity and index , then there exists an invertible matrix such that is a diagonal matrix whose first diagonal entries equal to 1, the following entries equal to -1, and the following entries equal to zero. Use this to prove theorem 3.1.

We shall now apply the preceding ideas to the study of projective quadrics on .

Exercise 3.8 (00)   Let be a quadratic form on and let be the associated projective quadric. If is an invertible linear transformation of , the quadric associated to the quadratic form is the image of under the projective transformation induced by the inverse of .

The preceding exercise tells us that if and are on the same orbit of the action of , then the projective quadrics they define are projectively equivalent. The converse is almost true, but not quite: the forms and have different indices, but they define the same conic.

Exercise 3.9 (05)   Let and be two quadratic forms on such that their associated quadrics are projectively equivalent. Show that the nullity of equals the nullity of and the index of is either equal to the index of or to minus the index of .

Exercise 3.10 (15)   Let be a nondegenerate quadratic form on and let be its associated quadric. Relate the index of to the maximum of the dimensions of all projective subspaces contained in .

Next: Conics on the real Up: The Projective Geometry of Previous: Basic algebraic concepts
Juan Carlos Alvarez 2001-01-30