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Symmetric Multilinear Forms#

Definition: Symmetric Multilinear Form

A multilinear form \(f: V^n \to F\) is symmetric if for each permutation \(\sigma\) of \(\{1,\dotsc, n\}\) and for all \(\mathbf{v}_1, \dotsc, \mathbf{v}_n \in V\), we have

\[f(\mathbf{v}_1, \dotsc, \mathbf{v}_n) = f(\mathbf{v}_{\sigma(1)}, \dotsc, \mathbf{v}_{\sigma(n)})\]

A multilinear form is symmetric if it remains the same under every rearrangement of its arguments.

Definiteness#

Definition: Definiteness

A symmetric multilinear form \(f: V^n \to F\) is:

  • positive definite if \(f(v, \dotsc, v) \gt 0\) for all \(v \in V \setminus \{0\}\);
  • positive semi-definite if \(f(v, \dotsc, v) \ge 0\) for all \(v \in V\);
  • negative definite if \(f(v, \dotsc, v) \lt 0\) for all \(v \in V \setminus \{0\}\);
  • negative semi-definite if \(f(v, \dotsc, v) \le 0\) for all \(v \in V\);
  • indefinite if there exist \(v, w \in V\) such that \(f(v, \dotsc, v) \gt 0\) and \(f(w, \dotsc, w) \lt 0\).