Definition 1 of the positive-definiteness note studied the case where is always non-negative (or positive). This note completes the picture by classifying every possible sign behaviour a quadratic form can exhibit — not just the always-positive case, but always-negative, mixed-sign, and the boundary semi-definite cases too — using the eigenvalue signs of throughout, and states the principal axes theorem that gives the geometric picture behind the algebra.
Learning objectives. By the end of this note you should be able to:
- define a quadratic form and write it in matrix notation;
- classify a quadratic form (definite, indefinite, semi-definite) from its matrix's eigenvalue signs;
- state the principal axes theorem and explain what it says geometrically;
- diagonalise a quadratic form via a change of variables.
In this lesson
- Quadratic forms
- The principal axes theorem
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.