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** Up:** The Projective Geometry of
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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.

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