What are eigenvalues and eigenvectors?

For a square matrix A, an eigenvector v is a nonzero vector that only gets scaled (not rotated) when multiplied by A: Av = λv. The scalar λ is the corresponding eigenvalue — it tells you by how much v is scaled. λ > 0 means same direction; λ 1 means stretched; 0 < |λ| < 1 means compressed; λ = 0 means collapsed to zero.

How do you find eigenvalues?

Solve the characteristic equation det(A − λI) = 0. Expanding this determinant gives the characteristic polynomial in λ. For a 2×2 matrix, it's a quadratic: λ² − tr(A)λ + det(A) = 0. For 3×3, it's a cubic. The roots of this polynomial are the eigenvalues. Then for each eigenvalue λ, solve (A − λI)v = 0 to find the eigenvectors.

What is the characteristic polynomial?

The characteristic polynomial of a matrix A is p(λ) = det(A − λI). For a 2×2 matrix: p(λ) = λ² − tr(A)λ + det(A). Its roots are the eigenvalues. The degree equals the matrix size. By the Cayley-Hamilton theorem, every matrix satisfies its own characteristic polynomial: p(A) = 0.

What do eigenvalues mean in real applications?

Eigenvalues reveal the fundamental behavior of a linear system. In PCA, eigenvalues of the covariance matrix are the variances along each principal direction. In Google PageRank, the ranking is the principal eigenvector of the link matrix. In physics, eigenvalues of the Hamiltonian are allowed energy levels. In vibration analysis, eigenvalues are the natural frequencies of oscillation.

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Q: How do you find eigenvalues?

Solve the characteristic equation det(A − λI) = 0. Expanding this determinant gives the characteristic polynomial in λ. For a 2×2 matrix, it's a quadratic: λ² − tr(A)λ + det(A) = 0. For 3×3, it's a cubic. The roots of this polynomial are the eigenvalues. Then for each eigenvalue λ, solve (A − λI)v = 0 to find the eigenvectors.

Q: What is the characteristic polynomial?

The characteristic polynomial of a matrix A is p(λ) = det(A − λI). For a 2×2 matrix: p(λ) = λ² − tr(A)λ + det(A). Its roots are the eigenvalues. The degree equals the matrix size. By the Cayley-Hamilton theorem, every matrix satisfies its own characteristic polynomial: p(A) = 0.

Q: What do eigenvalues mean in real applications?

Eigenvalues reveal the fundamental behavior of a linear system. In PCA, eigenvalues of the covariance matrix are the variances along each principal direction. In Google PageRank, the ranking is the principal eigenvector of the link matrix. In physics, eigenvalues of the Hamiltonian are allowed energy levels. In vibration analysis, eigenvalues are the natural frequencies of oscillation.

Eigenvalue Calculator

Find eigenvalues, eigenvectors, and the characteristic polynomial of 2×2 and 3×3 matrices. Shows full derivation including the characteristic equation and discriminant analysis.

Quick Answer

Av = λv defines eigenvalue λ and eigenvector v · Find λ by solving det(A−λI) = 0 (characteristic polynomial) · Then solve (A−λI)v = 0 for each λ · Eigenvalues can be real or complex.

Matrix Size

Eigenvalue λ	Effect on eigenvector	Example
λ = 1	No change — eigenvector is fixed point	Identity matrix
λ > 1	Stretched in eigenvector direction	Scaling transformation
0 < λ < 1	Compressed toward origin	Contraction
λ = 0	Collapsed to zero vector	Matrix is singular
λ < 0	Reversed direction and scaled	Reflection + scale
λ = a ± bi	Rotation + scaling (complex pair)	Rotation matrix

Eigenvalues by Application

Principal Component Analysis (PCA): The eigenvalues of the data covariance matrix are the variances explained by each principal component. Eigenvectors give the principal directions. This is why PCA "finds the most important axes" in your data.
Google PageRank: Web pages are ranked by the principal eigenvector (λ₁) of the link matrix — pages linked to by important pages are themselves important.
Vibration Analysis: Structural eigenvalues are the squared natural frequencies. Each mode shape (eigenvector) tells engineers how the structure deforms during resonance.
Quantum Mechanics: Eigenvalues of the Hamiltonian operator are the allowed energy levels. Eigenstates are stationary states where only the phase changes over time.
Differential Equations: For x' = Ax, the solution involves eˡᵗv where λ is the eigenvalue and v the eigenvector. Complex eigenvalues give oscillatory solutions.

Frequently Asked Questions

What is the Cayley-Hamilton theorem?

Every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A−λI) = λ² − tr(A)λ + det(A), then p(A) = A² − tr(A)A + det(A)I = 0 (the zero matrix). This theorem is used to compute powers of matrices and matrix functions without explicitly computing each power.

Can two eigenvectors share the same eigenvalue?

Yes — if λ has multiplicity > 1 (a repeated root of the characteristic polynomial), there may be multiple linearly independent eigenvectors for that λ. The set of all eigenvectors for λ (plus the zero vector) forms the eigenspace for λ. Its dimension is the geometric multiplicity of λ, which can be less than or equal to the algebraic multiplicity (how many times λ appears as a root).

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