MA22020: Advanced linear algebra

5.2 Symmetric bilinear forms

Definition. A bilinear form \(B:V\times V\to \F \) is symmetric if, for all \(v,w\in V\),
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} B(v,w)=B(w,v) \end{equation*}

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
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} \rad B:=\set {v\in V\st \text {$B(v,w)=0$, for all $w\in V$}}. \end{equation*}

We shall shortly see that \(\rad B\leq V\).

We say that \(B\) is non-degenerate if \(\rad B=\set 0\).

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 \(B\colon V\times V\to \F \) be a symmetric bilinear form with matrix \(A\) with respect to a basis \(\lst {v}1n\). Then \(v=\sum _{i=1}^nx_iv_i\in \rad B\) if and only if \(A\bx =0\) if and only if \(\bx ^TA=0\).

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
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} \rank B=\rank A. \end{equation*}

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 \(\half =(1+1)^{-1}\) 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 \(\lst {v}1n\) of \(V\) with respect to which the matrix of \(B\) is diagonal:
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} B(v_i,v_j)=0, \end{equation*}

for all \(\bw 1{i\neq j}n\). We call \(\lst {v}1n\) 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 \(P^TAP\) 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 \(v\in V\setminus \set 0\).

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
\(\seteqnumber{0}{5.}{0}\)
\begin{align*} p&=\max \set {\dim U\st \text {$U\leq V$ with $B_{|U\times U}$ positive definite}}\\ q&=\max \set {\dim W\st \text {$W\leq V$ with $B_{|W\times W}$ negative definite}}. \end{align*}

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!).