Action of $GL(n;{\Bbb R})$ on the space of quadratic forms on ${\Bbb R}^n$

Let $A$ be a symmetric $n \times n$ matrix and $Q(v) := v^{t}Av$ its associated quadratic form. If $M$ is an $n \times n$ invertible matrix, then $Q(Mv) = v^tM^tAMv$. Notice that the function $v \mapsto Q(Mv)$ 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 $\mu(A,M) := M^tAM$ defines a right action of $GL(n,{\Bbb R})$ on the (vector) space of $n \times n$ symmetric matrices.

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

Definition 3.1   Let $Q$ be a quadratic form on ${\Bbb R}^n$. The nullity of $Q$ is the nullity of its associated symmetric bilinear form. The index of $Q$ is $m$ if there exists an $m$-dimensional subspace $W$ such that the restriction of $Q$ to $W$ is negative definite, and there exist no higher dimensional subspaces with this property.

Exercise 3.2 (05)   Let $Q$ be a quadratic form on ${\Bbb R}^n$, let $M$ be an invertible $n \times n$ matrix, and let $Q_{M}$ be the quadratic form $v \mapsto Q(Mv)$. Show that the nullity and the index of $Q$ and $Q_{M}$ are equal.

Exercise 3.3 (05)   Show that if $A$ is a diagonal $n \times n$ 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 $Q$ and $Q'$ be quadratic forms on ${\Bbb R}^n$. If the nullity and the index of $Q$ and $Q'$ are equal, then there exists an invertible matrix $M$ such that $Q'(v) = Q(Mv)$ for all $v \in {\Bbb R}^n$.

In particular, the set of orbits of the action of the general linear group $GL(n;{\Bbb R})$ on the space of quadratic forms on ${\Bbb R}^n$ is in one-to-one correspondence with the set of pairs $(\nu,i)$, where $\nu$ and $i$ are nonnegative integers with $\nu + i \leq n$.

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 $A$ is an $n \times n$ symmetric matrix, then there exists an orthogonal matrix $R$ such that $R^{-1}AR$ is diagonal.

Notice that if $R$ is orthogonal, $R^{-1} = R^t$. This means that every orbit contains a diagonal matrix.

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

Exercise 3.6 (05)   Let $D$ be an $n \times n$ diagonal matrix such that $\nu$ entries in the diagonal are equal to zero, $i$ entries are equal to -1, and the rest are equal to 1. Show that there exists an invertible matrix $P$ such that $P^tDP$ is a diagonal matrix whose first $n - i - \nu$ diagonal entries equal to 1, the following $i$ entries equal to -1, and the following $\nu$ entries equal to zero. Hint: take $P$ to be a permutation matrix.

Exercise 3.7 (05)   Show that if $A$ is an $n \times n$ symmetric matrix whose associated quadratic form has nullity $\nu$ and index $i$, then there exists an invertible matrix $M$ such that $M^tAM$ is a diagonal matrix whose first $n - i - \nu$ diagonal entries equal to 1, the following $i$ entries equal to -1, and the following $\nu$ 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 ${\Bbb R}P^n$.

Exercise 3.8 (00)   Let $Q$ be a quadratic form on ${\Bbb R}^{n+1}$ and let ${\cal Q} \subset {\Bbb R}P^n$ be the associated projective quadric. If $M$ is an invertible linear transformation of ${\Bbb R}^{n+1}$, the quadric associated to the quadratic form $Q \circ M$ is the image of ${\cal Q}$ under the projective transformation induced by the inverse of $M$.

The preceding exercise tells us that if $Q$ and $Q'$ are on the same orbit of the action of $GL(n+1;{\Bbb R})$, then the projective quadrics they define are projectively equivalent. The converse is almost true, but not quite: the forms $x_{1}^{2} + x_{2}^2 - x_{3}^2$ and $-x_{1}^2 - x_{2}^2 + x_{3}^2$ have different indices, but they define the same conic.

Exercise 3.9 (05)   Let $Q$ and $Q'$ be two quadratic forms on ${\Bbb R}^{n+1}$ such that their associated quadrics are projectively equivalent. Show that the nullity of $Q$ equals the nullity of $Q'$ and the index of $Q$ is either equal to the index of $Q'$ or to $n+1$ minus the index of $Q'$.

Exercise 3.10 (15)   Let $Q$ be a nondegenerate quadratic form on ${\Bbb R}^{n+1}$ and let ${\cal Q} \subset {\Bbb R}P^n$ be its associated quadric. Relate the index of $Q$ to the maximum of the dimensions of all projective subspaces contained in ${\cal Q}$.