5.3 Application: Quadratic forms

  • Convention. We continue working with a field \(\F \) where \(1+1\neq 0\).

  • Definition. A quadratic form on a vector space \(V\) over \(\F \) is a function \(Q:V\to \F \) of the form

    \begin{equation*} Q(v)=B(v,v), \end{equation*}

    for all \(v\in V\), where \(B:V\times V\to \F \) is a symmetric bilinear form.

  • Lemma 5.9. Let \(Q:V\to \F \) be a quadratic form with \(Q(v)=B(v,v)\) for a symmetric bilinear form \(B\). Then

    \begin{equation*} B(v,w)=\half \bigl (Q(v+w)-Q(v)-Q(w)\bigr ), \end{equation*}

    for all \(v,w\in V\).

    \(B\) is called the polarisation of \(Q\).

  • Definitions. Let \(Q\) be a quadratic form on a finite-dimensional vector space \(V\) over \(\F \).

    The rank of \(Q\) is the rank of its polarisation.

    If \(\F =\R \), the signature of \(Q\) is the signature of its polarisation.

  • Theorem 5.10. Let \(Q\) be a quadratic form with rank \(r\) polarisation on a finite-dimensional vector space over \(\F \).

    • (1) When \(\F =\C \), there is a basis \(\lst {v}1n\) of \(V\) such that

      \begin{equation*} Q(\sum _{i=1}^nx_iv_i)=\plst {x^2}1r. \end{equation*}

    • (2) When \(\F =\R \) and \(Q\) has signature \((p,q)\), there is a basis \(\lst {v}1n\) of \(V\) such that

      \begin{equation*} Q(\sum _{i=1}^nx_iv_i)=\plst {x^2}1p-x_{p+1}^2-\dots -x_r^2. \end{equation*}