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Eigenvalue and Generalized Eigenvalue Problems: Tutorial

arXiv 1903.11240 ↗First posted 25 Mar 2019stat.ML cs.LGcited by 45

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Abstract (arXiv)

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also provide examples from machine learning, including principal component analysis, kernel supervised principal component analysis, and Fisher discriminant analysis, which result in eigenvalue and generalized eigenvalue problems. Finally, we introduce the solutions to both eigenvalue and generalized eigenvalue problems.

Key facts. Cited by 45 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Mar 2019.

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Citing papers per year: 2021: 7; 2022: 7; 2023: 11; 2024: 8; 2025: 10; 2026: 2.

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  1. 2021Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Surveycited by 36
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  10. 2025Position: Curvature Matrices Should Be Democratized via Linear Operatorscited by 9
  11. 2021Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Surveycited by 8
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  9. 2022Rayleigh EigenDirections (REDs): GAN latent space traversals for multidimensional featurescited by 2
  10. 2023Zoo Guide to Network Embeddingcited by 2
  11. 2024Self-consistent Validation for Machine Learning Electronic Structurecited by 2
  12. 2025Kernel VICReg for Self-Supervised Learning in Reproducing Kernel Hilbert Spacecited by 2
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  15. 2021Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Surveycited by 1
  16. 2021Sufficient Dimension Reduction for High-Dimensional Regression and Low-Dimensional Embedding: Tutorial and Surveycited by 1
  17. 2022Gravitational Dimensionality Reduction Using Newtonian Gravity and Einstein's General Relativitycited by 1
  18. 2023Scatter-based common spatial patterns -- a unified spatial filtering frameworkcited by 1
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  21. 2024Quantifying Public Response to COVID-19 Events: Introducing the Community Sentiment and Engagement Indexcited by 1
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  23. 2026Diagnosing and Repairing Unsafe Channels in Vision-Language Models via Causal Discovery and Dual-Modal Safety Subspace Projectioncited by 1
  24. 2022Application of Common Spatial Patterns in Gravitational Waves Detectioncited by 0
  25. 2022A Generalized EigenGame with Extensions to Multiview Representation Learningcited by 0
  26. 2023Towards Understanding Gradient Approximation in Equality Constrained Deep Declarative Networkscited by 0
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  30. 2025Next Token Perception Score: Analytical Assessment of your LLM Perception Skillscited by 0
  31. 2025Can multimodal representation learning by alignment preserve modality-specific information?cited by 0
  32. 2025A Unified Framework for EEG Seizure Detection Using Universum-Integrated Generalized Eigenvalues Proximal Support Vector Machinecited by 0
  33. 2025EEG-based Graph-guided Domain Adaptation for Robust Cross-Session Emotion Recognitioncited by 0