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Vectors that preserve direction under a linear transformation, scaled by associated eigenvalues.
DESCRIPTION
Eigenvectors define invariant directions of matrices and linear transformations. In PCA, principal components are eigenvectors of the covariance matrix. Orthogonal eigenvectors enable diagonalization of symmetric operators. Algorithms like power iteration estimate dominant eigenvectors for large problems. Applications span dimensionality reduction, image compression, and graph algorithms.
ALTERNATIVE DEFINITIONS
Wikipedia
An eigenvector of a linear transformation is a nonzero vector that changes by only a scalar factor when that linear transformation is applied to it.
Focuses on direction preservation under transformation.
MathWorks
Eigenvalues and eigenvectors of a square matrix A satisfy the equation Ax = lambdax.
Operational definition used in numerical computation.
Khan Academy
Eigenvectors are special vectors whose direction doesn’t change after a transformation; eigenvalues are the factors by which they’re stretched.
Provides intuitive explanation with visual learning.
RELATED TERMS
Eigenvalues Linear Transformation Diagonalization Principal Component Analysis Singular Value Decomposition Spectral Clustering Basis Power Iteration Orthogonality Rayleigh Quotient
COMPETING TERMS
Singular Vectors
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer Data Analyst Software Developer
EXTERNAL SOURCES
- Wikipedia - Eigenvalues and eigenvectors
- NumPy - Numerical eigenvector computation
- MIT OpenCourseWare - Eigenvectors lecture materials
- SciPy - SciPy eig documentation
TOP QUESTIONS
Why are Eigenvectors central to PCA?
Principal components are eigenvectors of the covariance matrix, revealing directions of maximal variance.
Category: Relevancy
How do Eigenvectors differ from singular vectors?
Eigenvectors require square matrices; singular vectors arise from SVD for any matrix.
Category: Alternatives
What is a common Eigenvector mistake?
Ignoring feature scaling before PCA can distort eigenvector directions.
Category: Mistakes
How are Eigenvectors computed in large problems?
Use iterative methods like power iteration, Lanczos, or Arnoldi processes.
Category: Performance
How do Eigenvectors aid spectral clustering?
Eigenvectors of graph Laplacians embed nodes for clustering in low-dimensional space.
Category: Example
What risks arise when misinterpreting Eigenvectors?
Misleading directions can cause poor dimensionality reduction and spurious patterns.
Category: Risk
What is orthogonality’s role in Eigenvectors?
For symmetric matrices, eigenvectors are orthogonal, simplifying projections and decompositions.
Category: Overview
What is the biggest misunderstanding about Eigenvectors?
They don’t necessarily represent physical directions; they are mathematical invariants of transformations.
Category: Misunderstanding
Where to learn Eigenvectors?
Start with MIT linear algebra lectures and practice with NumPy/SciPy eig functions.
Category: Learning