Quadratic Forms
Expressions xᵀAx classified by eigenvalues.
Key idea
Expressions xᵀAx classified by eigenvalues.
In depth
A quadratic form is q(x) = xᵀAx, with A symmetric. Diagonalising A removes cross terms, revealing the shape: ellipse, hyperbola and so on. Eigenvalue signs classify it.
Worked example
x² + 4xy + y² has A = [1 2; 2 1], eigenvalues 3 and −1, so a hyperbola.
Real-world uses
- Describing ellipses and energy surfaces
- Classifying curves in computer graphics and physics
Common mistake
Writing the cross term coefficient as an off-diagonal entry without halving.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Quadratic Forms?
Expressions xᵀAx classified by eigenvalues. It is a topic at the expert level in Linear Algebra.
Where is Quadratic Forms used in real life?
Real-world examples: Describing ellipses and energy surfaces; Classifying curves in computer graphics and physics.
What should I learn before Quadratic Forms?
Make sure you are comfortable with Positive Definite Matrices, the previous topic in Linear Algebra.
What is a common mistake with Quadratic Forms?
Writing the cross term coefficient as an off-diagonal entry without halving.
How can I practise Quadratic Forms?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Quadratic Forms?
Move on to Singular Value Decomposition next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.