Project 10: Invariant Subspace and Spectral Decomposition Lab

Build a spectral-analysis lab that enforces theorem preconditions, computes invariant subspaces, and reports stability diagnostics for symmetric and non-normal cases.

Quick Reference

Attribute Value
Difficulty Expert (Level 4)
Time Estimate 2-3 weeks
Language Python (alternatives: MATLAB, Julia)
Prerequisites Eigen decomposition, orthogonality, operator classes
Key Topics Spectral theorem, invariant subspaces, Schur fallback

1. Learning Objectives

  1. Enforce spectral theorem assumptions before decomposition.
  2. Compute and compare invariant subspaces across matrix classes.
  3. Interpret spectral gaps and subspace stability.
  4. Distinguish eigenvector-level instability from subspace-level stability.
  5. Build diagnostics suitable for engineering decision-making.

2. Theoretical Foundation

2.1 Core Concepts

  • Invariant subspaces as decomposition units.
  • Spectral theorem for symmetric/Hermitian operators.
  • Subspace angles and projection residuals.
  • Schur decomposition for non-normal matrices.

2.2 Why This Matters

Spectral methods drive PCA, graph embeddings, vibration modes, and covariance modeling. Misapplying theorem assumptions causes silent interpretability and stability failures.

2.3 Historical Context / Background

Spectral theory evolved from solving linear systems to classifying operators by decomposition behavior. Modern numerical practice adds robust alternatives (Schur, Krylov methods) for non-ideal matrices.

2.4 Common Misconceptions

  • If eigenvalues exist, spectral theorem always applies.
  • Eigenvectors are uniquely defined outputs.
  • High eigenvalue always implies actionable signal.

3. Project Specification

3.1 What You Will Build

A lab CLI/notebook that:

  • Detects operator class (symmetric/Hermitian/normal/general)
  • Chooses valid decomposition path
  • Computes top-k invariant subspaces
  • Reports residuals, orthogonality, and principal-angle metrics

3.2 Functional Requirements

  1. Matrix class detector with tolerance policy.
  2. Structured decomposition pipeline (eigh vs schur).
  3. Subspace comparison utilities.
  4. Case-study runner for covariance and graph matrices.
  5. Report export with theorem-precondition traces.

3.3 Non-Functional Requirements

  • Reliability: never run invalid theorem path silently.
  • Interpretability: report should explain why path was chosen.
  • Reproducibility: deterministic random seeds and config snapshots.

3.4 Example Usage / Output

$ python spectral_lab.py --case covariance --topk 5
[INFO] operator_class=symmetric
[INFO] path=spectral_theorem
[INFO] eigenvalues=[15.2, 8.7, 4.4, 1.1, 0.6]
[INFO] orthogonality_residual=2.3e-09
[INFO] subspace_reconstruction_error=1.7e-04

3.5 Real World Outcome

You end with a reusable spectral diagnostics workflow that can be dropped into ML/physics pipelines to gate decomposition validity and monitor subspace drift.

4. Solution Architecture

4.1 High-Level Design

Input Matrix -> Classifier -> Decomposition Path -> Subspace Analyzer -> Reporter

4.2 Key Components

Component Responsibility Key Decisions
Classifier Determine operator assumptions tolerance + structure checks
Decomposer Compute eig/Schur outputs path-by-class policy
Subspace Analyzer principal angles, residuals robust metrics
Reporter narrative + numeric output theorem traceability

4.3 Data Structures

SpectralReport:
  operator_class
  decomposition_path
  eigen_or_schur_data
  residual_metrics
  subspace_metrics

4.4 Algorithm Overview

  1. Validate symmetry/normality assumptions.
  2. Run decomposition path consistent with class.
  3. Compute top-k invariant subspace.
  4. Measure reconstruction/orthogonality/subspace angles.
  5. Export diagnostics and interpretations.

5. Implementation Guide

5.1 Development Environment Setup

Install NumPy/SciPy/Matplotlib
Prepare deterministic matrix fixture set
Configure tolerance profile

5.2 Project Structure

P10/
  classify/
  decompose/
  subspaces/
  cases/
  reports/

5.3 The Core Question You’re Answering

“When is spectral decomposition mathematically valid, and how robust is the extracted subspace?”

5.4 Concepts You Must Understand First

  1. Spectral theorem assumptions
  2. Invariant subspaces and projections
  3. Subspace-angle interpretation

5.5 Questions to Guide Your Design

  1. What tolerance defines “effectively symmetric” for your domain?
  2. How do you communicate uncertainty when eigenvalues cluster?

5.6 Thinking Exercise

Construct a pair of matrices with similar eigenvalues but different normality. Predict which outputs will remain stable across perturbations.

5.7 The Interview Questions They’ll Ask

  1. Why does symmetry guarantee orthonormal eigenvectors?
  2. What is an invariant subspace operationally?
  3. Why compare subspaces instead of raw eigenvectors?
  4. What is the role of spectral gap?
  5. When is Schur decomposition preferable?

5.8 Hints in Layers

  • Hint 1: Gate every run by class detector.
  • Hint 2: Validate decomposition residual before interpretation.
  • Hint 3: Use principal angles for cross-run comparison.
  • Hint 4: Add non-normal stress fixtures early.

5.9 Books That Will Help

Topic Book Chapter
Spectral theorem Strang Eigen chapters
Numerical stability Trefethen & Bau Eigenvalue computation
Practical APIs SciPy docs eigh and schur

5.10 Implementation Phases

  1. Class detector + fixtures
  2. Decomposition pathways
  3. Subspace metrics
  4. Reporting + case studies

5.11 Key Implementation Decisions

  • Keep theorem precondition checks mandatory, not optional.
  • Report both theorem-level and numerical-level confidence metrics.

6. Testing Strategy

6.1 Test Categories

  • Symmetric/Hermitian fixtures
  • Non-normal fixtures
  • Nearly symmetric edge cases

6.2 Critical Test Cases

  1. SPD covariance matrix with clean spectral gap.
  2. Non-normal matrix where eigenvectors are ill-conditioned.
  3. Clustered eigenvalue case with subspace ambiguity.

6.3 Test Data

  • Synthetic matrices + real covariance/graph samples.

7. Common Pitfalls & Debugging

  • Symptom: Orthonormality check fails in symmetric path.
    • Why: Matrix is not symmetric within tolerance.
    • Fix: Tighten classification or data cleaning.
    • Quick test: monitor ||A-A^T||.
  • Symptom: Top-k vectors unstable run-to-run.
    • Why: Eigenvalue cluster rotates basis.
    • Fix: compare principal subspace, not vector identities.
    • Quick test: principal-angle report.

8. Extensions & Challenges

  • Add iterative methods (Lanczos/Arnoldi) for large sparse cases.
  • Integrate streaming subspace tracking for drift detection.
  • Add graph Laplacian clustering case study.

9. Real-World Connections

  • PCA pipelines in analytics/ML.
  • Modal decomposition in mechanical simulation.
  • Graph spectral embeddings and segmentation.

10. Resources

11. Self-Assessment Checklist

  • I enforce theorem assumptions before decomposition.
  • I can explain subspace-angle metrics and when to use them.
  • My reports distinguish structural vs numerical uncertainty.
  • I can justify fallback to Schur pathway.

12. Submission / Completion Criteria

  • Functional lab with class-aware decomposition routing.
  • Case-study report for at least three matrix families.
  • One reflection on failure cases and interpretation limits.

This guide expands Project 10 from LINEAR_ALGEBRA_LEARNING_PROJECTS.md.