Quadratic Forms and Symmetric Matrices
When Matrices Meet Geometry
Explore how symmetric matrices and their eigenvalues determine the shape of quadratic functions — from bowls to saddles. Build toward positive definiteness, the spectral theorem, and the geometry behind least squares.
Sections
Functions on Rⁿ
8 min · 2 quiz questions
Constant Functions
5 min · 2 quiz questions
Linear Functions
8 min · 2 quiz questions · Interactive diagram
Quadratic Functions
10 min · 2 quiz questions
Shapes of Quadratic Functions
10 min · 2 quiz questions · Interactive diagram
Symmetric Matrices
10 min · 2 quiz questions · Interactive diagram
Symmetry is Preserved Under Congruence
7 min · 2 quiz questions
The Spectral Theorem
12 min · 2 quiz questions · Interactive diagram
Freedom in the Spectral Decomposition
7 min · 2 quiz questions
Eigenvalues and Eigenvectors
12 min · 2 quiz questions · Interactive diagram
Invertibility and Eigenvalues
8 min · 2 quiz questions · Interactive diagram
The Spectral Decomposition Gives Eigenvalues
8 min · 2 quiz questions
Rank from the Spectral Decomposition
7 min · 2 quiz questions · Interactive diagram
The Principal Form of a Quadratic
10 min · 2 quiz questions · Interactive diagram
Signature
8 min · 2 quiz questions
Definiteness
12 min · 2 quiz questions · Interactive diagram
Definiteness–Signature Relationship
7 min · 2 quiz questions
Normal Equations Always Give a PSD Matrix
10 min · 2 quiz questions · Interactive diagram