Eigenvalues and Eigenvectors Step by Step
Eigenvalues and eigenvectors are central to linear algebra and appear on exams in introductory university courses through engineering programs. This set covers finding eigenvalues via the characteristic polynomial, computing eigenvectors for each value, and understanding geometric versus algebraic multiplicity — foundational for diagonalization, differential equations, and PCA.
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5 CardsGeometric vs algebraic multiplicity
When is a matrix diagonalizable?
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What is the difference between eigenvalues and eigenvectors?
Eigenvalues are scalars (λ) satisfying det(A − λI) = 0 that scale an eigenvector. Eigenvectors are nonzero vectors whose direction is unchanged under Av = λv.
- Eigenvalues are found first via the characteristic polynomial
- Eigenvectors are computed from each eigenvalue
How many eigenvalues does an n×n matrix have?
An n×n matrix has exactly n eigenvalues counting algebraic multiplicity, though they may be repeated or complex. A 3×3 matrix has exactly 3 eigenvalues.
Why are eigenvalues important in data science?
Eigenvalues underpin Principal Component Analysis (PCA), which reduces high-dimensional data by projecting onto eigenvectors of the covariance matrix with the largest eigenvalues. They also appear in Google's PageRank and Markov chains.