Interactive Linear Algebra
Dan Margalit & Joseph Rabinoff · Georgia Tech · Open textbook
Read §§5.1 and 5.4 for eigenvectors and diagonalization.
Open the sourceEigenvectors and the sign of a geometry
A matrix can reveal directions, not just mix coordinates.
Prerequisites: 08 · A matrix is an action, 10 · Inner products and projection
Understand
Objectives
- Verify two eigenpairs by direct multiplication.
- Compute a symmetric quadratic form and report its three-entry signature.
- Distinguish zero vector, a nonzero null vector of Q, and a zero eigenvalue.
Start with something you can see
Apply a matrix to several arrows. Most turn; an eigenvector stays on its own line, possibly reversing or collapsing to zero. Predict the result for e₁=(1,0) and e₂=(0,1) under A=diag(2,−1): one doubles and the other reverses. The MIT eigenvalue passage includes negative and zero multipliers at 1:40. [1]
Give the idea a precise name
For a matrix A:ℝ²→ℝ², a nonzero vector v is an eigenvector if Av=λv for a real number λ, its eigenvalue. Zero can be an eigenvalue even though the zero vector is never an eigenvector. Its quadratic form Q(v)=vᵀAv is a scalar, and its signature (positive, negative, zero) counts eigenvalues with each sign, including multiplicity.
Av=λv; Q(v)=vᵀAv
The transpose vᵀ turns the vector into a row, so vᵀAv is one real number. For an eigenvector, Q(v)=λ‖v‖², making its sign match λ. But Q(v)=0 for a nonzero vector need not make that vector an eigenvector or force λ=0: positive and negative contributions can cancel.
Every real symmetric matrix has an orthogonal basis of eigenvectors. Along each basis direction, the sign of Q matches the sign of that direction’s eigenvalue.
Work one small world
For the symmetric A=[[1,2],[2,1]], A(1,1)=3(1,1) and A(1,−1)=−(1,−1). Thus eigenvalues are 3 and −1 and the signature is (1,1,0). Its quadratic form is Q(x,y)=x²+4xy+y²: Q(1,1)=6 and Q(1,−1)=−2. The two eigenvectors are orthogonal.
The picture uses the simpler form Q(x,y)=x²−y² of D=diag(1,−1), which appears again in the counterexample below.
| Q for diag(1,−1) | Vector | Value |
|---|---|---|
| Positive direction | (1,0) | 1 |
| Negative direction | (0,1) | −1 |
| Nonzero null vector | (1,1) | 0 |
Counterexample to “Q(v)=0 means A has a zero eigenvalue”: D=diag(1,−1) has eigenvalues 1 and −1, neither zero, but its nonzero vector (1,1) has Q(1,1)=0. This null vector of the quadratic form is not the zero vector and is not in the kernel of D. By contrast diag(2,0) really has a zero eigenvalue with nonzero eigenvector (0,1).
THE BRIDGEThe later Chow-ring degree pairing has the same matrix [[1,2],[2,1]]; its one positive and one negative direction will become numerically inspectable.
WHERE THIS IDEA STOPSOrthogonal diagonalization holds for real symmetric matrices, not every real matrix. An indefinite quadratic form is not a squared Euclidean norm.
Experiment
Compute A(1,1), A(1,−1) and their Q-values. Then sketch the sign regions of x²−y²; use the table to verify the diagonals carry nonzero null vectors.
What changes when a matrix has one positive direction and all remaining directions negative?
Check & explain
For A=[[2,4],[4,2]], what is the eigenvalue of (1,−1)?
For D=diag(2,−2), why can Q(2,2)=0?
- D must have eigenvalue zero
- Positive and negative contributions cancel for a nonzero vector
- The zero vector is an eigenvector
Teach back
Find the eigenpairs and signature of [[1,2],[2,1]], then use diag(1,−1) to explain why a nonzero null vector of Q need not signal a zero eigenvalue.
- Verify each matrix-vector multiplication.
- Use the (positive,negative,zero) signature order.
- Keep Q(v)=0 distinct from Av=0 and v=0.
Read deeper
Mathematics for Machine Learning
Deisenroth, Faisal & Ong · Freely readable textbook
Read §§3.2.3, 4.2 and 4.4, printed pp.105 and 115 for the eigenvalue sections.
Open the source21. Eigenvalues and Eigenvectors
Gilbert Strang · MIT OpenCourseWare · Lecture · 51 min
Start at 1:40. Watch the short core 1:40–4:40. The full 51-minute lecture is available from the same link.
Open the sourceEigenvectors and eigenvalues | Chapter 14, Essence of linear algebra
3Blue1Brown · Video · 17 min
Watch the 17-minute creator explanation of eigenvectors and eigenvalues.
Open the source