This capstone lesson — the Oxford M1 Linear Algebra II tradition's closing application — takes every result of Topic 5 and points it at the oldest question in the topic's ancestry: what does the equation actually look like? The answer, via the principal axes theorem, is that every such curve or surface is an ellipse, hyperbola, ellipsoid, or one of their relatives — rotated and scaled, but never anything more exotic — and the eigenvalues of read off exactly which one, with no case-by-case geometric reasoning required.
Learning objectives. By the end of this note you should be able to:
- write the general second-degree equation in matrix form;
- classify a conic (in 2D) using the eigenvalue signs of its matrix;
- classify a quadric surface (in 3D) using the same method;
- explain the connection between this note's classification and Topic 5's optimisation classification.
In this lesson
- Conics in matrix form
- Quadric surfaces in 3D
Before you start
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