Module 03 / §3.6–3.11
Understanding matrices
Explore eigenvectors, symmetry, and the geometry of quadratic forms.
8 lessons · Visual explanations · Guided experiments
03.1 / Explore → understand → practice
Which directions stay on the same line?
Identify eigenvectors and track eigenvalues under shifts, inverses, and powers.
03.2 / Explore → understand → practice
Which changes leave the diagonal sum alone?
Use trace linearity and cyclic invariance without assuming arbitrary reordering.
03.3 / Explore → understand → practice
When does a transformation flatten area?
Use the five determinant identities and distinguish area scale from length scale.
03.4 / Explore → understand → practice
Which matrices preserve length and angle?
Derive orthogonal matrix identities and prove preservation of inner products.
03.5 / Explore → understand → practice
Can a symmetric map be split into three simple steps?
Derive the spectral decomposition from an orthonormal eigenbasis.
03.6 / Explore → understand → practice
Which direction gives the largest quadratic value?
Prove the Rayleigh bounds and their equality conditions, including repeated eigenvalues.
03.7 / Explore → understand → practice
When is every nonzero direction uphill?
Prove the eigenvalue criteria and explain why a positive diagonal shift removes singularity.
03.8 / Explore → understand → practice
Why does a larger eigenvalue make a shorter ellipse axis?
Derive quadratic level sets from a sphere and identify their eigenvector directions.