Transform
Specify the matrix, coordinate order, units and starting vector.
SRAI Book 1 · Chapter 6 · PU-B01-C06
Eigenvalues, Eigenvectors and Spectral Intuition
Understand special directions, eigenpair residuals and repeated transformations. Connect spectral radius, power iteration and PCA to reproducible computation, while distinguishing numerical accuracy from model validity and decision relevance.
01 / LEARNING OUTCOMES
Understand what a transformation amplifies and what its results establish.
- Interpret eigenvalues and eigenvectors as multipliers and special directions.
- Verify an eigenpair using normalization and a scaled residual.
- Use an eigenvector basis to understand repeated transformations.
- Distinguish long-run decay from temporary amplification.
- Explain power iteration and its convergence limitations.
- Connect covariance eigenvectors to PCA without confusing variance with accuracy.
02 / ANALYTICAL SEQUENCE
Identify a direction, verify its multiplier, then interpret the result.
Identify
Find nonzero vectors satisfying Av = λv.
Verify
Normalize the vector and check the eigenpair residual against an explicit tolerance.
Interpret
Separate numerical accuracy, dynamical behaviour and decision relevance.
03 / GEOMETRY AND COMPUTATION
A dominant mode is a mathematical property, not a recommendation.
For the symmetric matrix with rows (2, 1) and (1, 2), the direction (1, 1) has multiplier 3, while (1, -1) has multiplier 1. Repetition amplifies these components differently.
A spectral radius below one guarantees eventual decay for every starting vector in a fixed finite-dimensional autonomous linear system. It does not rule out temporary growth.
CORE CONTROLS
| Eigenvector | Nonzero; normalization checked |
| Residual | Small relative to the declared scale |
| Power iteration | Convergence assumptions and residual checked |
| PCA | Centering, scaling and sample scope declared |
The notebook includes counterexamples and numerical assertions.
Explained variance is not predictive accuracy.
The synthetic PCA example describes variation in four observations, in the chosen units. It does not establish population representativeness or predictive performance.
Normalized sector weights are not budget shares.
The synthetic sector model illustrates an amplifying mode. Policy recommendations require validated relationships, objectives, costs, constraints and uncertainty analysis.
04 / REPRODUCIBILITY AND RELEASE CONTROL
Run the complete notebook with its verified runtime.
- The owner reported successful execution in Windows VS Code and Google Colab.
- The unchanged srai_math 1.1.1rc1 runtime is a disclosed prerelease accepted by the owner for this lesson. The bootstrap verifies its SHA-256.
- Visible lesson-level safeguards handle the shared runtime's zero-vector and power-iteration limitations. No blanket claim of independent runtime verification is made.
- Restart the kernel or Colab runtime and run all cells in order. Inspect the formulas and figures as well as the final checks.
- All PCA observations and sector coefficients are synthetic teaching examples. The model step is not calibrated as one year.
CONTROLLED RESOURCES
Read, reproduce, practise and review.
READ
View the complete GitHub production unit ↗
Controlled Chapter 6
Controlled publication edition.
PDF ↗ REPRODUCECanonical notebook
Execution-tested and owner-approved.
IPYNB ↗ PRACTISEExercises and solutions
Progress toward independent application.
PDF ↗ RUNOpen in Google Colab
Run the approved notebook with its checksum-pinned runtime.
COLAB ↗ DOWNLOADComplete release package
Chapter, notebook, exercises, Executive Brief, runtime and validation records.
ZIP ↓ APPLYExecutive brief
Connect analytical controls to decisions.
PDF ↗