Opening idea
Most vectors change direction when a matrix acts on them. For example, if $$ A= \begin{bmatrix} 2&1\\ 1&2 \end{bmatrix},
Big question. Directions that survive a transformation
Chapter 17 · Linear Algebra Computation
Eigenvalues and Eigenvectors
Directions that survive a transformation
Eigenvectors
Characteristic polynomial
Dynamics
Power iteration
Learning goals
- Define eigenvalues and eigenvectors.
- Check whether a vector is an eigenvector.
- Interpret eigenvectors geometrically.
- Derive the characteristic equation.
- Compute eigenvalues of $2\times 2$ matrices.
- Find eigenspaces by solving homogeneous systems.
- Understand algebraic and geometric multiplicity.
- Recognize when a matrix may be diagonalizable.
- Use Python to compute and visualize eigenvectors.
- Connect eigenvalues to dynamics, stability, and data.
Core formulas
Eigenvector equation
$$Av=\lambda v,\qquad v\ne 0$$
Characteristic equation
$$\det(A-\lambda I)=0$$
Trace/determinant for 2 by 2
$$\lambda_1+\lambda_2=\operatorname{tr}(A),\quad \lambda_1\lambda_2=\det(A)$$
Diagonalization preview
$$A=SDS^{-1}\quad\Longrightarrow\quad A^k=SD^kS^{-1}$$
Chapter map from the uploaded QMD
The interactive page follows the chapter sequence and turns key computational ideas into live tools.
- 17.1 The eigenvector equation
- 17.2 Geometric meaning
- 17.3 First example
- 17.4 Checking an eigenvector
- 17.5 Python plot: eigenvectors versus ordinary vectors
- 17.6 From eigenvector equation to homogeneous system
- 17.7 Characteristic polynomial
- 17.8 Example: eigenvalues of a 2\times 2 matrix
- 17.9 Finding eigenspaces
- 17.10 Example: eigenspaces
- 17.11 Python: eigenvalues and eigenvectors
- 17.12 Triangular matrices
- 17.13 Example: triangular matrix
- 17.14 Trace and determinant
- 17.15 Complex eigenvalues
- 17.16 Eigenvalues and powers of a matrix
Interactive Lab 1: Eigenvalues of a $2\times2$ matrix
Adjust $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. Real eigen-directions are shown when they exist.
Eigen report
Interactive Lab 2: Eigenvector checker
Check whether $v$ is an eigenvector by comparing $Av$ with $v$.
Interactive Lab 3: Dynamics $x_{k+1}=Ax_k$
Algorithmic summary
- 1Compute $\det(A-\lambda I)$.
- 2Solve the characteristic equation for eigenvalues.
- 3For each $\lambda$, solve $(A-\lambda I)v=0$.
- 4Use eigenvectors to understand stretching, sign flip, collapse, or rotation.
- 5For dynamics, study repeated multiplication by following dominant directions.
Common mistakes to avoid
- Allowing the zero vector as an eigenvector.
- Solving $\det(A)=0$ instead of $\det(A-\lambda I)=0$.
- Finding eigenvalues but not eigenspaces.
- Assuming every matrix has real eigenvalues.
- Assuming repeated eigenvalues always give enough eigenvectors.
AI-age connection. Eigenvectors organize long-term behavior in graph ranking, PCA, spectral clustering, Markov chains, stability analysis, and dimensionality reduction.
Quick checks
What is the main computational object in this chapter?
Use the formulas and live labs above: identify the matrix/vector operation that turns the concept into computation.
What geometric picture should you keep in mind?
Follow the canvas visualization: a vector, subspace, transformed square, fitted line, or special direction depending on the chapter.
What should be checked after computation?
Check the defining property: orthogonality, $Q^TQ=I$, residual condition, determinant sign/size, or $Av=\lambda v$.