MA22020: Advanced linear algebra

Chapter 5 Symmetric bilinear forms and quadratic forms

5.1 Bilinear forms and matrices

Definition. Let \(V\) be a vector space over a field \(\F \). A map \(B:V\times V\to \F \) is bilinear if it is linear in each slot separately:
\(\seteqnumber{0}{5.}{0}\)
\begin{align*} B(\lambda v_1+v_2,v)&=\lambda B(v_1,v)+B(v_2,v)\\ B(v,\lambda v_1+v_2)&=\lambda B(v,v_1)+B(v,v_2), \end{align*} for all \(v,v_1,v_2\in V\), \(v,v_1,v_2\in V\) and \(\lambda \in \F \).

A bilinear map \(V\times V\to \F \) is called a bilinear form on \(V\).

Definition. Let \(V\) be a vector space over \(\F \) with basis \(\cB =\lst {v}1n\) and let \(B:V\times V\to \F \) be a bilinear form. The matrix of \(B\) with respect to \(\cB \) is \(A\in M_{n}(\F )\) given by
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} A_{ij}=B(v_i,v_j), \end{equation*}

for \(\bw 1{i,j}n\).

Proposition 5.1. Let \(B:V\times V\to \F \) be a bilinear form with matrix \(A\) with respect to \(\cB =\lst {v}1n\). Then \(B\) is completely determined by \(A\): if \(v=\sum _{i=1}^nx_iv_i\) and \(w=\sum _{j=1}^ny_jv_j\) then
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} B(v,w)=\sum _{i,j=1}^nx_iy_jA_{ij}=\bx ^TA\by . \end{equation*}

Proposition 5.2. Let \(B:V\times V\to \F \) be a bilinear form with matrices \(A\) and \(A'\) with respect to bases \(\cB :\lst {v}1n\) and \(\cB ':\lst {v'}1n\) of \(V\). Then
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} A'=P^TAP \end{equation*}

where \(P\) is the change of basis matrix¹from \(\cB \) to \(\cB '\): thus \(v'_j=\sum _{i=1}^nP_{ij}v_i\), for \(\bw 1jn\).

¹ Algebra 1B, Definition 1.6.1.

Definition. We say that matrices \(A,B\in M_n(\F )\) are congruent if there is \(P\in \GL (n,\F )\) such that
\(\seteqnumber{0}{5.}{0}\)
\begin{equation*} B=P^TAP. \end{equation*}