Linear Algebra › Section G · Symmetric matrices and quadratic forms › Topic G.2
Quadratic forms
Energy, variance, squared distance and the curvature of a surface at a critical point are all quadratic expressions in several variables. Writing them with a symmetric matrix lets the spectral theorem strip out the cross terms and reveal their shape at a glance.
In this topic
- What a quadratic form is
- Reading off the matrix
- Removing cross terms by a change of variable
- Worked example: an ellipse in disguise
- The geometry of level curves
- Classifying quadratic forms
- Worked example: classifying two forms
- Shortcuts and the Cholesky factorisation
- Where definiteness matters
- Common mistakes to avoid
- Key takeaways
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