MA22020: Advanced linear algebra

5.2 Symmetric bilinear forms

Definition. A bilinear form $B : V \times V \to F$ is symmetric if, for all $v, w \in V$ ,

$B (v, w) = B (w, v)$

5.2.1 Rank and radical

Definitions. Let $B : V \times V \to F$ be a symmetric bilinear form.

The radical $rad B$ of $B$ is given by

$rad B := {v \in V | B (v, w) = 0, for all w \in V} .$

We shall shortly see that $rad B \leq V$ .

We say that $B$ is non-degenerate if $rad B = {0}$ .

If $V$ is finite-dimensional, the rank of $B$ is $\dim V - \dim rad B$ (so that $B$ is non-degenerate if and only if $rank B = \dim V$ ).

Lemma 5.3. Let $B : V \times V \to F$ be a symmetric bilinear form with matrix $A$ with respect to a basis $v_{1}, \dots, v_{n}$ . Then $v = \sum_{i = 1}^{n} x_{i} v_{i} \in rad B$ if and only if $A x = 0$ if and only if $x^{T} A = 0$ .

Corollary 5.4. Let $B : V \times V \to F$ be a symmetric bilinear form on a finite-dimensional vector space $V$ with matrix $A$ with respect to some basis of $V$ . Then

$rank B = rank A .$

In particular, $B$ is non-degenerate if and only if $det A \neq 0$ .

5.2.2 Classification of symmetric bilinear forms

Convention. In this section, we work with a field $F$ where $1 + 1 \neq 0$ so that $\frac{1}{2} = (1 + 1)^{- 1}$ makes sense. This excludes, for example, the $2$ -element field $Z_{2}$ .

Lemma 5.5. Let $B : V \times V \to F$ be a symmetric bilinear form such that $B (v, v) = 0$ , for all $v \in V$ . Then $B \equiv 0$ .

Theorem 5.6 (Diagonalisation Theorem). Let $B$ be a symmetric bilinear form on a finite-dimensional vector space over $F$ . Then there is a basis $v_{1}, \dots, v_{n}$ of $V$ with respect to which the matrix of $B$ is diagonal:

$B (v_{i}, v_{j}) = 0,$

for all $1 \leq i \neq j \leq n$ . We call $v_{1}, \dots, v_{n}$ a diagonalising basis for $B$ .

Corollary 5.7. Let $A \in M_{n \times n} (F)$ be symmetric. Then there is an invertible matrix $P \in GL (n, F)$ such that $P^{T} A P$ is diagonal.

5.2.3 Sylvester’s Theorem

Definitions. Let $B$ be a symmetric bilinear form on a real vector space $V$ .

Say that $B$ is positive definite if $B (v, v) > 0$ , for all $v \in V ∖ {0}$ .

Say that $B$ is negative definite if $- B$ is positive definite.

If $V$ is finite-dimensional, the signature of $B$ is the pair $(p, q)$ where

$\begin{aligned} p & = max {\dim U | U \leq V with B_{| U \times U} positive definite} \\ q & = max {\dim W | W \leq V with B_{| W \times W} negative definite} . \end{aligned}$

Theorem 5.8 (Sylvester’s Law of Inertia). Let $B$ be a symmetric bilinear form of signature $(p, q)$ on a finite-dimensional real vector space Then:
- • $p + q = rank B$ ;
- • any diagonal matrix representing $B$ has $p$ positive entries and $q$ negative entries (necessarily on the diagonal!).